Chapter 5

Use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100.200\), and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=\sin x \quad \text { on } \quad[0, \pi]$$

Use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100.200\), and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=\sin ^{2} x \quad \text { on } \quad[0, \pi]$$

Find the area between the curves \(y=\ln x\) and \(y=\ln 2 x\) from \(x=1\) to \(x=5\).

Use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100.200\), and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=x \sin \frac{1}{x} \quad \text { on } \quad\left[\frac{\pi}{4}, \pi\right]$$

Find the area between the curve \(y=\tan x\) and the \(x\) -axis from \(x=-\pi / 4\) to \(x=\pi / 3\).

Use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100.200\), and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=x \sin ^{2} \frac{1}{\pi} \quad \text { on } \quad\left[\frac{\pi}{4}, \pi\right]$$

Find the area of the "triangular" region in the first quadrant that is bounded above by the curve \(y=e^{2 x}\), below by the curve \(y=e^{x}\) and on the right by the line \(x=\ln 3\).

Use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100.200\), and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=x e^{-x} \quad \text { on } \quad[0,1]$$

Find the area of the "triangular" region in the first quadrant that is bounded above by the curve \(y=e^{x / 2},\) below by the curve \(y=e^{-x / 2},\) and on the right by the line \(x=2 \ln 2\).

Use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100.200\), and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=e^{-x^{2}} \quad \text { on } \quad[0,1]$$

Find the area of the region between the curve \(y=2 x /\left(1+x^{2}\right)\) and the interval \(-2 \leq x \leq 2\) of the \(x\) -axis.

Use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100.200\), and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=\frac{\ln x}{x} \quad \text { on } \quad[2,5]$$

Find the area of the region between the curve \(y=2^{1-x}\) and the interval \(-1 \leq x \leq 1\) of the \(x\) -axis.

Use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100.200\), and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=\frac{1}{\sqrt{1-x^{2}}} \text { on }\left[0, \frac{1}{2}\right]$$

Find the area of the region between the curve \(y=3-x^{2}\) and the line \(y=-1\) by integrating with respect to a. \(x,\) b. \(y\).

Find the area of the region in the first quadrant bounded on the left by the \(y\) -axis, below by the line \(y=x / 4,\) above left by the curve \(y=1+\sqrt{x},\) and above right by the curve \(y=2 / \sqrt{x}\).

Suppose the area of the region between the graph of a positive continuous function \(f\) and the \(x\) -axis from \(x=a\) to \(x=b\) is 4 square units. Find the area between the curves \(y=f(x)\) and \(y=2 f(x)\) from \(x=a\) to \(x=b\).

True, sometimes true, or never true? The area of the region between the graphs of the continuous functions \(y=f(x)\) and \(y=g(x)\) and the vertical lines \(x=a\) and \(x=b(a

Suppose that \(F(x)\) is an antiderivative of \(f(x)=(\sin x) / x\) \(x>0 .\) Express. $$\int_{1}^{3} \frac{\sin 2 x}{x} d x$$ in terms of \(F\)

Show that if \(f\) is continuous, then $$\int_{0}^{1} f(x) d x=\int_{0}^{1} f(1-x) d x$$

Suppose that $$\int_{0}^{1} f(x) d x=3$$ Find $$\int_{-1}^{0} f(x) d x$$ if a. \(f\) is odd, \(\quad\) b. \(f\) is even.

a. Show that if \(f\) is odd on \([-a, a],\) then \(x\) $$\int_{-a}^{a} f(x) d x=0$$ b. Test the result in part (a) with \(f(x)=\sin x\) and \(a=\pi / 2\)

If \(f\) is a continuous function, find the value of the integral $$I=\int_{0}^{a} \frac{f(x) d x}{f(x)+f(a-x)}$$ by making the substitution \(u=a-x\) and adding the resulting integral to \(I\)

By using a substitution, prove that for all positive numbers \(x\) and \(y\) $$\int_{x}^{x y} \frac{1}{t} d t=\int_{1}^{y} \frac{1}{t} d t$$

A basic property of definite integrals is their invariance under translation, as expressed by the equation $$\begin{aligned}&\int_{a}^{b} f(x) d x=\int_{a-c}^{b-c} f(x+c) d x\\\ &(1)\end{aligned}$$ The equation holds whenever \(f\) is integrable and defined for the necessary values of \(x .\) For example in the accompanying figure, show that $$\int_{-2}^{-1}(x+2)^{3} d x=\int_{0}^{1} x^{3} d x$$ because the areas of the shaded regions are congruent. (GRAPH CAN'T COPY). Use a substitution to verify Equation (1)

A basic property of definite integrals is their invariance under translation, as expressed by the equation $$\begin{aligned}&\int_{a}^{b} f(x) d x=\int_{a-c}^{b-c} f(x+c) d x\\\ &(1)\end{aligned}$$ The equation holds whenever \(f\) is integrable and defined for the necessary values of \(x .\) For example in the accompanying figure, show that $$\int_{-2}^{-1}(x+2)^{3} d x=\int_{0}^{1} x^{3} d x$$ because the areas of the shaded regions are congruent. (GRAPH CAN'T COPY). For each of the following functions, graph \(f(x)\) over \([a, b]\) and \(f(x+c)\) over \([a-c, b-c]\) to convince yourself that Equation (1) is reasonable. a. \(f(x)=x^{2}, \quad a=0, \quad b=1, \quad c=1\) b. \(f(x)=\sin x, \quad a=0, \quad b=\pi, \quad c=\pi / 2\) c. \(f(x)=\sqrt{x-4}, \quad a=4, \quad b=8, \quad c=5\)

In Exercises \(119-122,\) you will find the area between curves in the plane when you cannot find their points of intersection using simple algebra. Use a CAS to perform the following steps: a. Plot the curves together to see what they look like and how many points of intersection they have. b. Use the numerical equation solver in your CAS to find all the points of intersection. c. Integrate \(|f(x)-g(x)|\) over consecutive pairs of intersection values. d. Sum together the integrals found in part (c). $$f(x)=\frac{x^{3}}{3}-\frac{x^{2}}{2}-2 x+\frac{1}{3}, \quad g(x)=x-1$$

In Exercises \(119-122,\) you will find the area between curves in the plane when you cannot find their points of intersection using simple algebra. Use a CAS to perform the following steps: a. Plot the curves together to see what they look like and how many points of intersection they have. b. Use the numerical equation solver in your CAS to find all the points of intersection. c. Integrate \(|f(x)-g(x)|\) over consecutive pairs of intersection values. d. Sum together the integrals found in part (c). $$f(x)=\frac{x^{4}}{2}-3 x^{3}+10, \quad g(x)=8-12 x$$

In Exercises \(119-122,\) you will find the area between curves in the plane when you cannot find their points of intersection using simple algebra. Use a CAS to perform the following steps: a. Plot the curves together to see what they look like and how many points of intersection they have. b. Use the numerical equation solver in your CAS to find all the points of intersection. c. Integrate \(|f(x)-g(x)|\) over consecutive pairs of intersection values. d. Sum together the integrals found in part (c). $$f(x)=x+\sin (2 x), \quad g(x)=x^{3}$$

In Exercises \(119-122,\) you will find the area between curves in the plane when you cannot find their points of intersection using simple algebra. Use a CAS to perform the following steps: a. Plot the curves together to see what they look like and how many points of intersection they have. b. Use the numerical equation solver in your CAS to find all the points of intersection. c. Integrate \(|f(x)-g(x)|\) over consecutive pairs of intersection values. d. Sum together the integrals found in part (c). $$f(x)=x^{2} \cos x, \quad g(x)=x^{3}-x$$