Show that for any constant a, the function y = sinh(ax)satisfies the equation y = a2y

Describe the method of slicing for finding volumes, and use that method to derive an integral formula for finding volumes by the method of disks.

State an integral formula for finding a volume by the method of cylindrical shells, and use Riemann sums to derive the formula.

State an integral formula for finding the arc length of a smooth curve \(y=f(x)\) over an interval \([a, b]\), and use Riemann sums to derive the formula. Equation Transcription: [a,b] Text Transcription: y=f(x) [a,b]

State an integral formula for the work W done by a variable force \(F(x)\) applied in the direction of motion to an object moving from \(x=a \text { to } x=b\), and use Riemann sums to derive the formula. Equation Transcription: Text Transcription: F(x) x=a x=b

State an integral formula for the fluid force F exerted on a vertical flat surface immersed in a fluid of weight density , and use Riemann sums to derive the formula.

Let R be the region in the first quadrant enclosed by \(y=x^{2}, y=2+x_{1} \text { and } x=0\). In each part, set up, but do not evaluate, an integral or a sum of integrals that will solve the problem. (a) Find the area of R by integrating with respect to x. (b) Find the area of R by integrating with respect to y. (c) Find the volume of the solid generated by revolving R about the x-axis by integrating with respect to x. (d) Find the volume of the solid generated by revolving R about the x-axis by integrating with respect to y. (e) Find the volume of the solid generated by revolving R about the y-axis by integrating with respect to x. (f ) Find the volume of the solid generated by revolving R about the y-axis by integrating with respect to y. (g) Find the volume of the solid generated by revolving R about the line \(y=-3\) by integrating with respect to x. (h) Find the volume of the solid generated by revolving R about the line \(x=5\) by integrating with respect to x. Equation Transcription: Text Transcription: y=x^2 y=2+x x=0 y=-3 x=5

(a) Set up a sum of definite integrals that represents the total shaded area between the curves \(y=f(x) \text { and } y=g(x)\) in the accompanying figure on the next page. (b) Find the total area enclosed between \(y=x^{3} \text { and } y=x \text { over the interval }[-1,2]\). Equation Transcription: [-1,2] Text Transcription: y=f(x) y=g(x) y=x^3 y=x [-1,2]

The accompanying figure shows velocity versus time curves for two cars that move along a straight track, accelerating from rest at a common starting line. (a) How far apart are the cars after 60 seconds? (b) How far apart are the cars after T seconds, where \(0 \leq T \leq 60\)? Equation Transcription: Text Transcription: 0 leq T leq 60

Let R be the region enclosed by the curves \(y=x^{2}+4 y=x^{3}\), and the y-axis. Find and evaluate a definite integral that represents the volume of the solid generated by revolving R about the x-axis. Equation Transcription: Text Transcription: y=x^2+4, y=x^3

A football has the shape of the solid generated by revolving the region bounded between the x-axis and the parabola y = 4R(x2 ? L2)/L2 about the x-axis. Find its volume. Equation Transcription: Text Transcription: y=4R(x^2-1/4L^2)/L^2

Find the volume of the solid whose base is the region bounded between the curves \(y=\sqrt{x} \text { and } y=1 / \sqrt{x} \text { for } 1 \leq x \leq 4\)and whose cross sections perpendicular to the x-axis are squares. Equation Transcription: Text Transcription: y=square root x y= 1/ square root x 1 leq x leq 4

Consider the region enclosed by y = sin?1 x, y = 0, and x = 1. Set up, but do not evaluate, an integral that represents the volume of the solid generated by revolving the region about the x-axis using (a) disks (b) cylindrical shells. Equation Transcription: Text Transcription: y=sin^-1, y=0 x=1

Find the arc length in the second quadrant of the curve\(x^{2 / 3}+y^{2 / 3}=4 \text { from } x=-8 \text { to } x=-1\). Equation Transcription: Text Transcription: x^?+y^? = 4 x= - 8 x= - 1

Let C be the curve\(y=e^{x} \text { between } x=0 \text { and } x=\ln 10\). In each part, set up, but do not evaluate, an integral that solves the problem. (a) Find the arc length of C by integrating with respect to x. (b) Find the arc length of C by integrating with respect to y. Equation Transcription: Text Transcription: y=e^x x=0 x=In 10

Find the area of the surface generated by revolving the curve \(y=\sqrt{25-x}, 9 \leq x \leq 16\) about the x-axis. Equation Transcription: Text Transcription: y= square root 25-x 9 leq x leq 16

Let C be the curve\(27 x-y^{3}=0 \text { between } y=0 \text { and } y=2\). In each part, set up, but do not evaluate, an integral or a sum of integrals that solves the problem. (a) Find the area of the surface generated by revolving C about the x-axis by integrating with respect to x. (b) Find the area of the surface generated by revolving C about the y-axis by integrating with respect to y. (c) Find the area of the surface generated by revolving C about the line \(y=-2\)by integrating with respect to y. Equation Transcription: Text Transcription: 27x-y^3=0 y=0 y=2 y=-2

Consider the solid generated by revolving the region enclosed by\(y=\sec x, x=0, x=\pi / 3\) about the x-axis. Find the average value of the area of a cross section of this solid taken perpendicular to the x-axis. Equation Transcription: Text transcription: y=sec x x=0 x=pi/3 y=0

Consider the solid generated by revolving the region enclosed by \(y=\sqrt{a^{2}-x^{2}} \text { and } \mathrm{y}=0\) about the x-axis. Without performing an integration, find the average value of the area of a cross section of this solid taken perpendicular to the x-axis. Equation Transcription: Text Transcription: y= square root a^2-x^2 y=0

(a) A spring exerts a force of 0.5 N when stretched 0.25 m beyond its natural length. Assuming that Hookes law applies, how much work was performed in stretching the spring to this length? (b) How far beyond its natural length can the spring be stretched with 25 J of work?

A boat is anchored so that the anchor is 150 ft below the surface of the water. In the water, the anchor weighs 2000 lb and the chain weighs \(30 \mathrm{lb} / \mathrm{ft}\). How much work is required to raise the anchor to the surface? Equation Transcription: Text Transcription: 30lb/ft

Find the centroid of the region. The region bounded by \(y^{2}=4 x \text { and } y^{2}=8(x-2)\). Equation Transcription: Text Transcription: y^2=4x y^2 = 8(x-2)

Find the centroid of the region. The upper half of the ellipse \((x / a)^{2}+(y b)^{2}=1\). Equation Transcription: Text Transcription: (x/a)^2+(y/b)^2 =1

In each part, set up, but do not evaluate, an integral that solves the problem. (a) Find the fluid force exerted on a side of a box that has a 3 m square base and is filled to a depth of 1 m with a liquid of weight density \(\mathrm{pN} / \mathrm{m}^{3}\). (b) Find the fluid force exerted by a liquid of weight density \(p l b / f t^{3}\) on a face of the vertical plate shown in part (a) of the accompanying figure. (c) Find the fluid force exerted on the parabolic dam in part (b) of the accompanying figure by water that extends to the top of the dam. Equation Transcription: Text Transcription: Rho N/m^3 Rho lb/ft^3

Show that for any constant \(a\), the function \(y=\sinh (a x)\)satisfies the equation \(y^{\prime \prime}=a^{2} y\). Equation Transcription: Text Transcription: a y=sinh(ax) y”=a^2y

In each part, prove the identity. \(\text { (a) } \cosh 3 x=4 \cosh ^{3} x-3 \cosh x\) \(\text { (b) } \cosh \frac{1}{2} x=\sqrt{\frac{1}{2}(\cosh x+1)}\) \(\text { (c) } \sinh \frac{1}{2} x=\pm \sqrt{\frac{1}{2}(\cosh x-1)}\) Equation Transcription: Text Transcription: Cosh 3x=4 cosh^3 x-3 cosh x Cosh 1/2 x = square root ½ (cosh x +1) Sinh ½ x = ph square root ½ (cosh x -1)