Areas of plane regions and other geometric applications of definite integrals.
These practice problems use a definite integral as a model. You translate a geometric or physical description into an integral, choose the limits, and then evaluate. A first course in definite integrals is assumed. The difficulty is usually the setup: which slice, washer, shell, or strip fits the solid or the region. Each solution shows that setup before the computation.
Areas covers plane regions bounded by given curves. Volumes and Geometry includes solids of revolution, solids with known cross-sections, arc length, and surface area, among them classical solids such as a torus. Physical Applications asks for work, mass, hydrostatic force, and centers of mass. Further Applications mixes those ideas in problems that still reduce to one definite integral.
If the antiderivative is what stalls you, return to Definite Integrals, the easier computational set, or to Integrals for the indefinite techniques. When the region is two-dimensional and the integrand depends on both variables, continue with Double Integrals. Work, mass, and motion stated in physical language also appear in Physics.
Problem 1
HardAreas
Find the area bounded by the curve $y = x^2 + x + 1$, the lines $x = 0$, $x = 1$, and $y = 0$.
Calculate the area bounded by the curve $y = \frac{1}{x}$, the lines $x = 1$, $x = a$, and $y = 0$, where $a > 0$. What does $P(a)$ approach as $a \to 0$?
Prove that the area bounded by the curve $y = kx^n$, the line $x = a$, and $y = 0$ equals $\frac{1}{n+1}$ times the area of the rectangle constructed over the coordinates of the point on the curve $y = kx^n$ with abscissa $a$.
Find the area bounded by the arc of the parabola $y^2 = 4a(a+x)$, the arc of the circle $x^2 + y^2 = 4a^2$, and the tangent to the circle at the point $(2a, 0)$.
At the intersection points of the line $x - y + 1 = 0$ and the parabola $y = x^2 - 4x + 5$, tangents are drawn to the parabola. Calculate the area bounded by the parabola and the tangents.
At the intersection points of the line $y = 0$ and the parabola $y = x^2 - 4$, normals are drawn to the parabola. Find the area bounded by the parabola and the normals.
Determine the coefficients $a, b, c,$ and $d$ such that the hyperbola $y = \frac{ax + b}{cx + d}$ passes through the origin and has asymptotes $x = 2$ and $y = 1$. Then calculate the area bounded by the arc of the resulting hyperbola and the line $x + y + 1 = 0$.
Find the area bounded by the curve $y = \ln\frac{\sqrt{x^2+1} - x}{\sqrt{x^2+1} + x}$ and the lines $x = 1$, $y = 0$. Based on this, prove the inequality: $\ln(\sqrt{2} - 1) < 1 - \sqrt{2}$.
From the point $\left(\frac{3}{5}, 0\right)$, draw tangents to the hyperbola $x^2 - y^2 = 1$ and find the area bounded by these tangents and the arc of the hyperbola between the tangent points.
Determine the parameter $m$ so that the curve $y^2 = mx$ is tangent to the curve $y = e^x$, and calculate the area bounded by these curves and the $y$-axis.
A parabola $y^2 = 2px$ intersects a circle $x^2 + y^2 - 2rx = 0$ at points $A$ and $B$. Determine the parameter $p$ such that the area bounded by the parabola and the chord $AB$ is maximized.
Find the volume generated by rotating the region bounded by the curves $y = \frac{1}{1 + x^2}$, $x = \pm 1$, and $y = 0$ about: 1° the $x$-axis, 2° the $y$-axis.
Find the volume generated by rotating the region bounded by the curve $g(x) = \int_0^x \frac{f(t^2)}{t^2} \, dt$, where $f(x) = \frac{x^{3/2}}{1 + x^2}$, and the lines $x = 0$, $x = 1$, $y = 0$ about the $x$-axis.
Find the volume of the solid generated by rotating the region bounded by the curves $y = e^x$, $x = 0$, and $y = 0$ about: 1° the $x$-axis, 2° the $y$-axis.
Find the volume of the solid generated by rotating the region bounded by the curves $y = a\sinh\frac{x}{a}$, $x = \pm a$, and $y = 0$ about the $x$-axis.
Find the volume of the solid generated by rotating the region bounded by the curves $y = x^2$ and $y^2 = x$ about: 1° the $x$-axis, 2° the line $x = y$.
The arc of the curve $xy^2 + x - y = 0$ between the middle inflection point and the point with the largest abscissa rotates about the $y$-axis. Find the volume of the resulting solid.
Find the volume of the solid generated by rotating one arch of the cycloid $x = a(t - \sin t)$, $y = a(1 - \cos t)$ about: 1° the $x$-axis, 2° the $y$-axis, 3° the line $x = a\pi$.
A glass in the shape of a circular cylinder of radius $r$ contains water. The axis of the glass is tilted at angle $\alpha$ to the horizontal. A segment of the base with central angle $2\varphi$ is covered by water. Find the volume of the water.
The center of a square moves along a diameter of a circle of radius $a$, with the plane of the square remaining perpendicular to the plane of the circle, and two opposite vertices of the square moving on the circle. Find the volume of the solid generated by the moving square.
Calculate the arc length of the curve $x = 3\left(\cos t + \ln\tan\frac{t}{2}\right)$, $y = 3\sin t$ between the points corresponding to parameter values $t_1 = \frac{\pi}{2}$ and $t_2 = \frac{2\pi}{3}$.
Prove that the arc length of the curve $y_1 = f(x)$ equals the arc length of the curve $y_2 = f(x) + c$ between points with abscissas $x = a$ and $x = b$. Give a geometric interpretation.
Prove that the arc length of the curve $y = f(x)$ between points with abscissas $x = a$ and $x = b$ equals the arc length of the curve $y = f(x - h)$ between points with abscissas $x = a + h$ and $x = b + h$. Give a geometric interpretation.
The line $x = t$ (with $t > 0$) meets the curves $y = a\sinh\dfrac{x}{a}$ and $y = a\cosh\dfrac{x}{a}$ at points $M_1$ and $M_2$ respectively, and meets the $x$-axis at point $M$. Show that the length of segment $MM_1$ equals the arc length of $y = a\cosh\dfrac{x}{a}$ from $T(0,a)$ to $M_2$.
Find the surface area generated by rotating one arch of the cycloid $x=a(t-\sin t),\ y=a(1-\cos t)$: a) about the $x$-axis, b) about the line $x=\pi a$.
Calculate the surface area generated by rotating the arc of the catenary $y = a\cosh\dfrac{x}{a}$ about the $x$-axis between $x=0$ and $x=a$ (where $a>0$).
Calculate the surface area generated by rotating the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ (with $a>b$) about: a) the $x$-axis (prolate spheroid), b) the $y$-axis (oblate spheroid).
Calculate the surface area generated by rotating the curve $y=4-\dfrac{\sqrt[8]{9x^2}}2$ about the $x$-axis for $x>0$ (the arc between the coordinate axes).
Determine the moment of inertia of a homogeneous solid cone with base radius $R$, height $h$, density $\mu$, and mass $M$, with respect to its base plane.
Prove that the static moment of an arbitrary arc of a parabola with respect to its axis is proportional to the difference of the radii of curvature at the endpoints of that arc, with proportionality coefficient $\dfrac p3$, where $p$ is the parabola's parameter.
Find the centroid of the solid generated by rotating about the $y$-axis the area bounded by $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ and the lines $y=0$ and $y=b$ ($b>0$).
Find the centroid of the solid generated by rotating about the $x$-axis the region bounded by $y=\cosh x$, $x=0$, $x=a$ (where $a>0$), and the $x$-axis.
A body is thrown vertically upward with initial velocity $v_0$, and (neglecting air resistance) its velocity at time $t$ is $v=v_0-gt$. Find its height above the launch point $t$ seconds after launch.
A body is thrown vertically upward with initial velocity $v_0$. Taking air resistance into account, its velocity is $v=c\tan\left(-\dfrac{gt}{c}+\arctan\dfrac{v_0}{c}\right)$, where $c$ is a constant. Find the maximum height reached.
A point moves harmonically along the $x$-axis with velocity $v=\dfrac{2\pi}{T}\cos\left(\dfrac{2\pi t}{T}+\varphi_0\right)$, where $T$ is the period and $\varphi_0$ the initial phase. Given that the point is at $x=x_1$ at time $t=t_1$, find its position at time $t_2$.
A point mass $m$ lies on the axis of a circular disk of surface density $\mu$, at distance $a$ from the disk's center. The distance from the point to the edge of the disk is $R$. Find the attractive force between the disk and the point.
Find the force of attraction between a homogeneous sphere of density $\mu=1$, radius $R$, and a material point of mass $m$ located at distance $b$ from the center of the sphere, where $b>R$.
A body falls freely toward the Earth from rest at distance $r_0$ from the Earth's center, in vacuum, taking into account that gravitational force varies with distance ($F=GMm/r^2$). Find the velocity $v$ as a function of the current distance $r$, and the time $t$ elapsed as a function of $r$.
A boiler has the shape of a hemisphere of radius $R$, sitting on a flat circular base with a domed top, and is completely filled with water. How much work is required to pump the water out through an opening at the top of the dome?
A ball of radius $R$ floats fully submerged in water (the specific weight of both the ball and the water is $1$). How much work is required to pull it completely out of the water?
Calculate the work required to lift a mass $m$ from the Earth's surface to height $h$, taking into account that gravitational force decreases with distance from the Earth's center. What is the work when $h\to\infty$?
Calculate the work required to pump water out of a boiler shaped like a paraboloid of revolution, with depth $H=0.5$ m and base (opening) radius $R=0.4$ m, given that the boiler is completely full.
A vertical axle carrying a load $P$ ends in a hemisphere of radius $R$ that bears against a socket, and the axle rotates. Assuming the normal pressure is spread uniformly over the hemispherical contact surface and the coefficient of friction is $\mu$, find the work done by friction in one complete revolution.
How much work must be done to build a conical pile of sand with base radius $R=1.2$ m and height $h=1$ m, if all the sand is lifted from ground level and the specific gravity of sand is $2$?
Determine the work expended when driving a nail to depth $d$ into a homogeneous material, given that the resistance offered by the material at depth $x$ is proportional to $x$.
A tank has the shape of an elliptic cylinder lying on its side: the base is an ellipse with horizontal semi-axis $a$ and vertical semi-axis $b$, and the cylinder's length is $L$. The tank is filled halfway (by volume) with liquid of specific gravity $\gamma=1$. Find the work needed to pump all the liquid out over the highest point of the tank.
Water drains from a filled cylindrical vessel of base area $S=420\ \text{cm}^2$ and height $H=40$ cm through a hole of area $s=2\ \text{cm}^2$ at the bottom. In what time will the vessel empty?
Air of initial volume $V_0=0.1\ \text{m}^3$ in a cylinder is under pressure $P_0=10330\ \text{kgf/m}^2$ from a piston. Determine the work of isothermal compression of the air to volume $V_1=0.03\ \text{m}^3$.
Two electric charges $e_0=100$ CGSE and $e_1=200$ CGSE lie on the $x$-axis at $x_0=0$ and $x_1=1$ cm. How much work is done by the electric field if the second charge moves to point $x_2=10$ cm?
Black-body radiation has spectral energy density $E_\lambda=\dfrac{c}{\lambda^5\left(e^{c'/(\lambda T)}-1\right)}$ at absolute temperature $T$, where $c,c'$ are constants. Find the total radiated energy $E=\int_0^\infty E_\lambda\,d\lambda$ as a function of $T$.
Calculate the kinetic energy of a uniform disk of mass $M$ and radius $R$ rotating with angular velocity $\omega$ about the axis through its center, perpendicular to its plane.
Calculate the kinetic energy of a uniform rectangular plate with sides $a$ and $b$ ($a>b$), rotating with constant angular velocity $\omega$ about the axis through its center parallel to the longer side $a$.
A uniform rod $AB$ of cross-sectional area $S=4\ \text{cm}^2$, length $l=20$ cm, and density $\gamma=7.8\ \text{g/cm}^3$ rotates with angular velocity $\omega=10\pi\ \text{s}^{-1}$ about an axis through one end, perpendicular to the rod. Find its kinetic energy of rotation.
Calculate the kinetic energy of a homogeneous right circular cone of mass $M$, base radius $R$, and height $H$, rotating with angular velocity $\omega$ about its axis of symmetry.
Determine the water pressure force on a vertical triangular plate whose base $a$ lies at the water surface and whose apex points straight down, at depth (height) $h$.
Calculate the water pressure force on a vertical trapezoidal plate whose upper base $a=70$ m lies at the water surface, whose lower base is $b=50$ m, and whose height is $h=20$ m.
Determine the water pressure force on a vertical semicircular plate of radius $a$ whose diameter $2a$ lies at the water surface (the semicircle hangs downward into the water).
A right circular cone of base radius $R$ and height $H$ is submerged apex-down in water, with its flat base exactly at the water surface. Find the total (vertical) hydrostatic force on the cone.
A cylindrical tank with horizontal axis is filled to half with a liquid of specific gravity $0.9$. Find the pressure force of the liquid on each circular end, given the radius $R=2$ m.
Wind exerts a constant pressure $p$ (force per unit area) on a rectangular door of width $b$ and height $h$, hinged along one vertical edge. Find the moment of the wind-pressure force about the hinge line.
Find how the intensity of light decreases as it passes through a homogeneous transparent medium, given that the intensity lost in a thin layer is proportional to both the intensity entering that layer and the layer's thickness.
A vertical column of height $h$ carries a load $Q$ at its top. Find how the cross-sectional area must vary with position if, at every height, the cross-section is proportional to the total force (applied load plus weight of column above) acting there.
Approximate $\dfrac{\pi}{4}=\displaystyle\int_0^1\frac{dx}{1+x^2}$ using (a) the midpoint rule with $n=10$, (b) the trapezoidal rule with $n=10$, and (c) Simpson's rule with $n=2$; estimate the error in each case.
Calculate $\displaystyle\int_0^1\frac{dx}{1+x}$ by dividing $[0,1]$ into 8 subintervals using (a) the rectangle (midpoint) formula, (b) the trapezoidal formula, (c) Simpson's formula, and compare each with the exact value.
Calculate the approximate value of $\pi$ by applying Simpson's formula to $\displaystyle\int_0^1\frac{dx}{1+x^2}$, with $2n=10$, to an accuracy of $0.01$.
Calculate $\ln2=\displaystyle\int_1^2\frac{dx}{x}$ with accuracy $0.001$ using (a) the rectangle (midpoint) method, (b) the trapezoidal method, (c) Simpson's method — determining in each case how many subintervals are needed.
Given that $\displaystyle\int_0^{1/2}\frac{dx}{\sqrt{1-x^2}}=\frac{\pi}{6}$, apply Simpson's formula with $n=4$ to approximate this integral and hence $\pi$, and estimate the error.