Chapter 6

Suppose a cut is made through a solid object perpendicular to the \(x\) -axis at a particular point \(x .\) Explain the meaning of \(A(x)\)

Draw the graphs of two functions \(f\) and \(g\) that are continuous and intersect exactly twice on \((-\infty, \infty) .\) Explain how to use integration to find the area of the region bounded by the two curves.

Explain the meaning of position, displacement, and distance traveled as they apply to an object moving along a line.

Give two pieces of information that may be used to formulate an exponential growth or decay function.

Give a geometrical interpretation of the function \(\ln x=\int_{1}^{x} \frac{d t}{t}\)

Sketch the graphs of \(y=\cosh x, y=\sinh x\) and \(y=\tanh x\) (include asymptotes), and state whether each function is even, odd, or neither.

A frustum of a cone is generated by revolving the graph of \(y=4 x\) on the interval [2,6] about the \(x\) -axis. What is the area of the surface of the frustum?

Explain the steps required to find the length of a curve \(x=g(y)\) between \(y=c\) and \(y=d.\)

Fill in the blanks: A region \(R\) is revolved about the \(y\) -axis. The volume of the resulting solid could (in principle) be found by using the disk/washer method and integrating with respect to ___________ to using the shell method and integrating with respect to ___________.

Describe how a solid of revolution is generated.

Draw the graphs of two functions \(f\) and \(g\) that are continuous and intersect exactly three times on \((-\infty, \infty) .\) Explain how to use integration to find the area of the region bounded by the two curves.

Suppose the velocity of an object moving along a line is positive. Are position, displacement, and distance traveled equal? Explain.

How much work is required to move an object from \(x=0\) to \(x=5\) (measured in meters) in the presence of a constant force of \(5 \mathrm{N}\) acting along the \(x\) -axis?

Explain the meaning of doubling time.

Evaluate \(\int 4^{x} d x\)

What is the fundamental identity for hyperbolic functions?

Suppose \(f\) is positive and differentiable on \([a, b] .\) The curve \(y=f(x)\) on \([a, b]\) is revolved about the \(x\) -axis. Explain how to find the area of the surface that is generated.

Arc length calculations Find the arc length of the following curves on the given interval by integrating with respect to \(x\). \(y=2 x+1 ;[1,5]\) (Use calculus.)

Fill in the blanks: A region \(R\) is revolved about the \(x\) -axis. The volume of the resulting solid could (in principle) be found by using the disk/washer method and integrating with respect to ___________ or using the shell method and integrating with respect to __________.

The region bounded by the curves \(y=2 x\) and \(y=x^{2}\) is revolved about the \(x\) -axis. Give an integral for the volume of the solid that is generated.

Make a sketch to show a case in which the area bounded by two curves is most easily found by integrating with respect to \(x\)

Given the velocity function \(v\) of an object moving along a line, explain how definite integrals can be used to find the displacement of the object.

Why must integration be used to find the work done by a variable force?

Explain the meaning of half-life.

What is the inverse function of \(\ln x,\) and what are its domain and range?

How are the derivative formulas for the hyperbolic functions and the trigonometric functions alike? How are they different?

Suppose \(g\) is positive and differentiable on \([c, d] .\) The curve \(x=g(y)\) on \([c, d]\) is revolved about the \(y\) -axis. Explain how to find the area of the surface that is generated.

Arc length calculations Find the arc length of the following curves on the given interval by integrating with respect to \(x\). $$y=\frac{1}{2}\left(e^{x}+e^{-x}\right) ;[-\ln 2, \ln 2]$$

The region bounded by the curves \(y=2 x\) and \(y=x^{2}\) is revolved about the \(y\) -axis. Give an integral for the volume of the solid that is generated.

Explain how to use definite integrals to find the net change in a quantity, given the rate of change of that quantity.

Express \(3^{x}, x^{\pi},\) and \(x^{\sin x}\) using the base \(e\).

Express \(\sinh ^{-1} x\) in terms of logarithms.

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=3 x+4 \text { on }[0,6]$$

Arc length calculations Find the arc length of the following curves on the given interval by integrating with respect to \(x\). $$y=\frac{1}{3} x^{3 / 2} ;[0,60]$$

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(y\) -axis. $$y=x-x^{2}, y=0$$

Why is the disk method a special case of the general slicing method?

Given the rate of change of a quantity \(Q\) and its initial value \(Q(0)\) explain how to find the value of \(Q\) at a future time \(t \geq 0\).

How are the rate constant and the doubling time related?

Find the arc length of the following curves on the given interval by integrating with respect to \(x.\) $$y=3 \ln x-\frac{x^{2}}{24} ;[1,6]$$

Evaluate \(\frac{d}{d x}\left(3^{x}\right)\).

What is the domain of \(\operatorname{sech}^{-1} x ?\) How is \(\operatorname{sech}^{-1} x\) defined in terms of the inverse hyperbolic cosine?

How are the rate constant and the half-life related?

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=12-3 x \text { on }[1,3]$$

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(y\) -axis. $$y=-x^{2}+4 x+2, y=x^{2}-6 x+10$$