Chapter 5

Decide whether the given statement is true or false. Then justify your answer. \(\int_{a}^{b} f(x) d x \geq 0\), then \(f(x) \geq 0\) for all \(x\) in \([a, b]\)

Find the area of the region under the curve \(y=f(x)\) over the interval \([a, b]\). To do this, divide the interval \([a, b]\) into n equal subintervals, calculate the area of the corresponding circumscribed polygon, and then let \(n \rightarrow \infty\). $$ y=x^{3} ; a=0, b=1 $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{1} x \cos ^{3}\left(x^{2}\right) \sin \left(x^{2}\right) d x\)

Let \(f(x)=|\sin x| \sin (\cos x)\). (a) Is \(f\) even, odd, or neither? (b) Note that \(f\) is periodic. What is its period? (c) Evaluate the definite integral of \(f\) for each of the following intervals: \([0, \pi / 2],[-\pi / 2, \pi / 2],[0,3 \pi / 2],[-3 \pi / 2,3 \pi / 2]\), \([0,2 \pi],[\pi / 6,13 \pi / 6],[\pi / 6,4 \pi / 3],[13 \pi / 6,10 \pi / 3]\).

Decide whether the given statement is true or false. Then justify your answer. If \(\int_{a}^{b} f(x) d x=0\), then \(f(x)=0\) for all \(x\) in \([a, b]\).

Find the area of the region under the curve \(y=f(x)\) over the interval \([a, b]\). To do this, divide the interval \([a, b]\) into n equal subintervals, calculate the area of the corresponding circumscribed polygon, and then let \(n \rightarrow \infty\). $$ y=x^{3}+x ; a=0, b=1 $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{-\pi / 2}^{\pi / 2} x^{2} \sin ^{2}\left(x^{3}\right) \cos \left(x^{3}\right) d x\)

Decide whether the given statement is true or false. Then justify your answer. If \(f(x) \geq 0\) and \(\int_{a} f(x) d x=0\), then \(f(x)=0\) for all \(x\) in \([a, b]\).

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{1} \frac{1}{1+x^{2}} d x\)

Decide whether the given statement is true or false. Then justify your answer. If \(\int_{a}^{b} f(x) d x>\int_{a}^{b} g(x) d x\), then $$ \int_{a}^{b}[f(x)-g(x)] d x>0 $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{-1}^{1} x^{2} \cosh x^{3} d x\)

Prove the Symmetry Theorem for the case of odd functions.

Decide whether the given statement is true or false. Then justify your answer. If \(f\) and \(g\) are continuous and \(f(x)>g(x)\) for all \(x\) in \([a, b]\), then \(\left|\int_{a}^{b} f(x) d x\right|>\left|\int_{a}^{b} g(x) d x\right|\).

Let \(A_{a}^{b}\) denote the area under the curve \(y=x^{2}\) over the interval \([a, b]\). (a) Prove that \(A_{0}^{b}=b^{3} / 3\). Hint \(: \Delta x=b / n\), so \(x_{i}=i b / n\); use circumscribed polygons. (b) Show that \(A_{a}^{b}=b^{3} / 3-a^{3} / 3\). Assume that \(a \geq 0\).

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{-5}^{5} x \sinh x^{2} d x\)

The velocity of an object is \(v(t)=2-|t-2|\). Assuming that the object is at the origin at time 0 , find a formula for its position at time \(t\). (Hint: You will have to consider separately the intervals \(0 \leq t \leq 2\), and \(t>2\).) When, if ever, does the object return to the origin?

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{1}^{3} \frac{\ln x}{x} d x\) Hint: Let \(u=\ln x\)

The velocity of an object is $$ v(t)= \begin{cases}5 & \text { if } 0 \leq t \leq 100 \\ 6-t / 100 & \text { if } 100700\end{cases} $$ (a) Assuming that the object is at the origin at time 0 , find a formula for its position at time \(t(t \geq 0)\). (b) What is the farthest to the right of the origin that this object ever gets? (c) When, if ever, does the object return to the origin?

From Special Sum Formulas 1-4 you might guess that $$ 1^{m}+2^{m}+3^{m}+\cdots+n^{m}=\frac{n^{m+1}}{m+1}+C_{n} $$ where \(C_{n}\) is a polynomial in \(n\) of degree \(m\). Assume that this is true (which it is) and, for \(a \geq 0\), let \(A_{a}^{b}\left(x^{m}\right)\) be the area under the curve \(y=x^{m}\) over the interval \([a, b]\). (a) Prove that \(A_{0}^{b}\left(x^{m}\right)=\frac{b^{m+1}}{(m+1)}\). (b) Show that \(A_{a}^{b}\left(x^{m}\right)=\frac{b^{m+1}}{m+1}-\frac{a^{m+1}}{m+1}\).

Suppose that \(f^{\prime}\) is integrable and \(\left|f^{\prime}(x)\right| \leq M\) for all \(x\). Prove that \(|f(x)| \leq|f(a)|+M|x-a|\) for every \(a\).

Water leaks out of a 200 -gallon storage tank (initially full) at the rate \(V^{\prime}(t)=20-t\), where \(t\) is measured in hours and \(V\) in gallons. How much water leaked out between 10 and 20 hours? How long will it take the tank to drain completely?

Oil is leaking at the rate of \(V^{\prime}(t)=1-t / 110\) from a storage tank that is initially full of 55 gallons. How much leaks out during the first hour? During the tenth hour? How long until the entire tank is drained?

The mass, in kilograms, of a rod measured from the left endpoint to the point \(x\) meters away is \(m(x)=x+x^{2} / 8\). What is the density \(\delta(x)\) of the rod, measured in kilograms per meter? Assuming that the rod is 2 meters long, express the total mass of the rod in terms of its density.

In Problems 73-76, first recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus \(\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(\frac{3 i}{n}\right)^{2} \frac{3}{n}\)

In Problems 73-76, first recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus \(\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(\frac{2 i}{n}\right)^{3} \frac{2}{n}\)

In Problems 73-76, first recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus \(\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[\sin \left(\frac{\pi i}{n}\right)\right] \frac{\pi}{n}\)

In Problems 73-76, first recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus \(\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[1+\frac{2 i}{n}+\left(\frac{2 i}{n}\right)^{2}\right] \frac{2}{n}\)

Explain why \(\left(1 / n^{3}\right) \sum_{i=1}^{n} i^{2}\) should be a good approximation to \(\int_{0}^{1} x^{2} d x\) for large \(n\). Now calculate the summation expression for \(n=10\), and evaluate the integral by the Second Fundamental Theorem of Calculus. Compare their values.

Evaluate \(\int_{-2}^{4}(2[x]-3|x|) d x\).

Show that \(\frac{1}{2} x|x|\) is an antiderivative of \(|x|\), and use this fact to get a simple formula for \(\int_{a}^{b}|x| d x\).

Suppose that \(f\) is continuous on \([a, b]\). (a) Let \(G(x)=\int_{a}^{x} f(t) d t\). Show that \(G\) is continuous on \([a, b]\). (b) Let \(F(x)\) be any antiderivative of \(f\) on \([a, b]\). Show that \(F\) is continuous on \([a, b]\).

Give an example to show that the accumulation function \(G(x)=\int_{a}^{x} f(x) d x\) can be continuous even if \(f\) is not continuous.