Chapter 6

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. \( y = \sin^2 x \) , \( y = \sin^4 x \) , \( 0 \le x \le \pi \) ; about \( x = \frac{\pi}{2} \)

Evaluate the integral and interpret it as the area of a region. Sketch the region. \( \displaystyle \int_{0}^{\frac{\pi}{2}} \mid \sin x - \cos 2x \mid dx \)

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. \( y = x^3 \sin x \) , \( y = 0 \) , \( 0 \le x \le \pi \) ; about \( x = -1 \)

Evaluate the integral and interpret it as the area of a region. Sketch the region. \( \displaystyle \int_{-1}^1 \mid 3^x - 2^x \mid dx \)

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. \( y = -x^2 + 6x - 8 \) , \( y = 0 \) ; about the y-axis

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. \( y = \sin^2 x \) , \( y = 0 \) , \( 0 \le x \le \pi \) ; about \( y = -1 \)

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. \( y = -x^2 + 6x - 8 \) , \( y = 0 \) ; about the x-axis

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. \( y = x \) , \( y = xe^{1 - \frac{x}{2}} \) ; about \( y = 3 \)

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. \( y = \frac{x}{(x^2 + 1)^2} \) , \( y = x^5 - x \) , \( x \ge 0 \)

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. \( y^2 - x^2 = 1 \) , \( y = 2 \) ; about the x-axis

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. \( y = 3x^2 - 2x \) , \( y = x^3 - 3x + 4 \)

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. \( y^2 - x^2 = 1 \) , \( y = 2 \) ; about the y-axis

Each integral represents the volume of a solid. Describe the solid. \( \pi \displaystyle \int_{-1}^1 (1 -y^2)^2 dy \)

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. \( y = 1.3^x \) , \( y = 2 \sqrt{x} \)

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. \( x^2 + (y - 1)^2 = 1 \) ; about the y-axis

Graph the region between the curves and use your calculator to compute the area correct to five decimal places. \( y = \frac{2}{1 + x^4} \) , \( y = x^2 \)

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. \( x = (y - 3)^2 \) , \( x = 4 \) ; about \( y = 1 \)

Graph the region between the curves and use your calculator to compute the area correct to five decimal places. \( y = e^{1 - x^2} \) , \( y = x^4 \)

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. \( x = (y - 1)^2 \) , \( x - y = 1 \) ; about \( x = -1 \)

A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square centimeters, are 0, 18, 58, 79, 94, 106, 117, 128, 63, 39, and 0. Use the Midpoint Rule to estimate the volume of the liver.

A log 10 m long is cut at 1-meter intervals and its cross- sectional areas \( A \) (at a distance x from the end of the log) are listed in the table. Use the Midpoint Rule with \( n = 5 \) to estimate the volume of the log.

Use a computer algebra system to find the exact area enclosed by the curves \( y = x^5 - 6x^3 + 4x \) and \( y = x \).

(a) A model for the shape of the bird's egg is obtained by rotating about the x-axis the region under the graph of $$ f(x) = (ax^3 + bx^2 + cx + d) \sqrt{1 - x^2} $$ Use \( CAS \) to find the volume of such an egg. (b) For a red-throated loon, \( a = -0.06 \), \( b = 0.04 \), \( c = 0.1 \), and \( d = 0.54 \). Graph \( f \) and find the volume of an egg of this species.

Sketch the region in the xy-plane defined by the inequalities \( x - 2y^2 \ge 0 \) , \( 1 - x - \mid y \mid \ge 0 \) and find its area.

Find the volume of the described solid \( S \). A right circular cone with height \( h \) and base radius \( r \).

Find the volume of the described solid \( S \). A frustum of a right circular cone with height \( h \), lower base radius \( R \), and top radius \( r \).

Find the volume of the described solid \( S \). A cap of a sphere with radius \( r \) and height \( h \).

Find the volume of the described solid \( S \). A frustum of a pyramid with square base of side \( b \), square top of side \( a \), and height \( h \) What happens if \( a = b \)? What happens if \( a = 0 \)?

If the birth rate of a population is \( b(t) = 2200 e^{0.024t} \) people per year and the death rate is \( d(t) = 1460 e^{0.018t} \) people per year, find the area between these curves for \( 0 \le t \le 10 \). What does this area represent?

Find the volume of the described solid \( S \). A pyramid with height \( h \) and rectangular base with dimensions \( b \) and \( 2b \).

Find the volume of the described solid \( S \). A pyramid with height \( h \) and base an equilateral triangle with side \( a \) (a tetrahedron).

The rates at which rain fell, in inches per hour, in two different locations \( t \) hours after the start of a storm are given by \( f(t) = 0.73t^3 - 2t^2 + t + 0.6 \) and \( g(t) = 0.17t^2 - 0.5t + 1.1 \). Compute the area between the graphs for \( 0 \le t \le 2 \) and interpret your result in this context.

Find the volume of the described solid \( S \). A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 3 cm, 4 cm, and 5 cm.

Find the volume of the described solid \( S \). The base of \( S \) is a circular disk with radius \( r \). Parallel cross sections perpendicular to the base are squares.

Find the volume of the described solid \( S \). The base of \( S \) is an elliptical region with boundary curve \( 9x^2 + 4y^2 = 36 \). Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.

The curve with equation \( y^2 = x^2 (x + 3) \) is called Tschirnhausen's cubic. If you graph this curve you will see that part of the curve forms a loop. Find the area enclosed by the loop.

Find the volume of the described solid \( S \). The base of \( S \) is the triangular region with vertices \( (0, 0) \), \( (1, 0) \), and \( (0, 1) \). Cross-sections perpendicular to the y-axis are equilateral triangles.

Find the area of the region bounded by the parabola \( y = x^2 \), the tangent line to this parabola at \( (1, 1) \), and the x-axis.

Find the number \( b \) such that the line \( y = b \) divides the region bounded by the curves \( y = x^2 \) and \( y = 4 \) into two regions with equal area.

Find the volume of the described solid \( S \). The base of \( S \) is the region enclosed by the parabola \( y = 1 - x^2 \) and the x-axis. Cross-sections perpendicular to the y-axis are squares.

(a) Find the number a such that the line \( x = a \) bisects the area under the curve \( y = \frac{1}{x^2} \), \( 1 \le x \le 4 \).

Find the values of \( c \) such that the area of the region bounded by the parabolas \( y = x^2 - c^2 \) and \( y = c^2 - x^2 \) is 576.

Find the volume of the described solid \( S \). The base of \( S \) is the region enclosed by \( y = 2 - x^2 \) and the x-axis. Cross-sections perpendicular to the y-axis are quarter-circles.

Suppose that \( 0 < c < \frac{\pi}{2} \). For what value of \( c \) is the area of the region enclosed by the curves \( y = \cos x \), \( y = \cos (x - c) \), and \( x = 0 \) equal to the area of the region enclosed by the curves \( y = \cos (x - c) \), \( x = \pi \), and \( y = 0 \).

For what values of \( m \) do the line \( y = mx \) and the curve \( y = \frac{x}{(x^2 + 1)} \) enclose a region? Find the area of the region.

The base of \( S \) is a circular disk with radius \( r \). Parallel cross- sections perpendicular to the base are isosceles triangles with height \( h \) and unequal side in the base. (a) Set up an integral for the volume of \( S \). (b) By interpreting the integral as an area, find the volume of \( S \).

(a) Set up an integral for the volume of a solid torus (the donut-shaped solid shown in the figure) with radii \( r \) and \( R \). (b) By interpreting the integral as an area, find the volume of the torus.

Find the volume common to two circular cylinders, each with radius \( r \), if the axes of the cylinders intersect at right angles.

Find the volume common to two spheres, each with radius \( r \), if the center of each sphere lies on the surface of the other sphere.

A bowl is shaped like a hemisphere with diameter 30 cm. A heavy ball with diameter 10 cm is placed in the bowl and water is poured into the bowl to a depth of \( h \) centimeters. Find the volume of water in the bowl.