Chapter 7

\(\sum_{i=1}^{6}(3 i-2)\)

\(\int_{-1}^{3}\left(3 x^{2}+5 x-1\right) d x\)

\(\sum_{i=2}^{5} \frac{i}{i-1}\)

\(f(x)=\sqrt{25-x^{2}} ;[a, b]=[-4.5,3] ; k=3\)

\(\int_{0}^{4}\left(y^{3}-y^{2}+1\right) d y\)

\(\sum_{j=3}^{6} \frac{2}{j(j-2)}\)

\(\int_{-1}^{3} \frac{d y}{(y+2)^{3}}\)

\(\sum_{i=-2}^{3} 2^{i}\)

\(\int_{-2}^{0} 3 w \sqrt{4-w^{2}} d w\)

\(\sum_{i=0}^{3} \frac{1}{1+i^{2}}\)

\(f(x)=\left\\{\begin{array}{ll}1+x & \text { if }-4 \leq x \leq-2 \\ 2-x & \text { if }-2

\(\int_{1}^{3} \frac{x d x}{\left(3 x^{2}-1\right)^{3}}\)

\(\sum_{k=1}^{4} \frac{(-1)^{k+1}}{k}\)

\(\int_{0}^{2} x^{2} d x\)

\(\int_{0}^{1} \frac{\left(y^{2}+2 y\right) d y}{\sqrt[3]{y^{3}+3 y^{2}+4}}\)

\(\sum_{k=-2}^{3} \frac{k}{k+3}\)

\(\int_{2}^{4} x^{2} d x\)

\(\int_{1}^{2} x^{3} d x\)

\(\int_{0}^{15} \frac{w d w}{(1+w)^{3 / 4}}\)

\(\int_{-2}^{1} x^{4} d x\)

\(\int_{1}^{4}\left(x^{2}+4 x+5\right) d x\)

\(\int_{-2}^{5}|x-3| d x\)

\(\int_{0}^{4}\left(x^{2}+x-6\right) d x\)

\(\int_{-3}^{3} \sqrt{3+|x|} d x\)

\(\int_{-2}^{2}\left(x^{3}+1\right) d x\)

\(\int_{-1}^{2}\left(4 x^{3}-3 x^{2}\right) d x\)

\(\int_{1}^{4} \frac{x^{5}-x}{3 x^{3}} d x\)

Use the method of this section to find the area of an isosceles trapezoid whose bases have measures \(b_{1}\) and \(b_{2}\) and whose altitude has measure \(h\).

Bounded by the line \(y=2 x-1\), the \(x\) axis, and the lines \(x=1\) and \(x=5\).

Suppose a ball is dropped from rest and after \(t\) sec its velocity is \(v \mathrm{ft} / \mathrm{sec}\). Neglecting air resistance, express \(v\) in terms of \(t\) as \(v=f(t)\) and find the average value of \(f\) on \([0,2]\).

$$ \int_{1}^{2} \frac{x^{3}+2 x^{2}+x+2}{(x+1)^{2}} d x $$ (HINT: Divide the numerator by the denominator.)

The graph of \(y=4-|x|\) and the \(x\) axis from \(x=-4\) to \(x=4\) form a triangle. Use the method of this section to find the area of this triangle.

\int_{-3}^{2} \frac{3 x^{3}-24 x^{2}+48 x+5}{x^{2}-8 x+16} d x

\(\sum_{i=1}^{25} 2 i(i-1)\)

Find the average value of the function \(f\) defined by \(f(x)=\sqrt{16-x^{2}}\) on the interval \([-4,4] .\) Draw a figure. (HINT: Find the value of the definite integral by interpreting it as the measure of the area of a region enclosed by a semicircle.)

\(\int_{1}^{64}\left(\sqrt{t}-\frac{1}{\sqrt{t}}+\sqrt[3]{t}\right) d t\)

\(\sum_{i=1}^{20} 3 i\left(i^{2}+2\right)\)

Show that if \(f\) is continuous on \([-1,2]\), then $$ \int_{-1}^{2} f(x) d x+\int_{2}^{0} f(x) d x+\int_{0}^{1} f(x) d x+\int_{1}^{-1} f(x) d x=0 $$

Find the average value of the function \(f\) defined by \(f(x)=\sqrt{49-x^{2}}\) on the interval \([0,7]\). Draw a figure. (HINT: Find the value of the definite integral by interpreting it as the measure of the area of a region enclosed by a quarter-circle.)

\(\int_{0}^{1} \sqrt{x} \sqrt{1+x \sqrt{x}} d x\)

\(\sum_{i=1}^{n}\left(10^{i+1}-10^{i}\right)\)

Show that \(\int_{0}^{1} x d x \geq \int_{0}^{1} x^{2} d x\) but \(\int_{1}^{2} x d x \leq \int_{1}^{2} x^{2} d x\). Do not evaluate the definite integrals.

Show that the intermediate-value theorem guarantees that the equation \(x^{3}-4 x^{2}+x+3=0\) has a root between 1 and \(2 .\)

In Exercises 19 through 22 , use Theorem \(7.6 .1\) to find the indicated derivative. \(D_{x} \int_{0}^{x} \sqrt{4+t^{5}} d t\)

\(\sum_{k=1}^{n}\left(2^{k-1}-2^{k}\right)\)

If \(f\) is continuous on \([a, b]\), prove that $$ \begin{gathered} \left|\int_{a}^{b} f(x) d x\right| \leq \int_{a}^{b}|f(x)| d x \\ (\text { HINT: }-|f(x)| \leq f(x) \leq|f(x)| .) \end{gathered} $$

Suppose \(f\) is a function for which \(0 \leq f(x) \leq 1\) if \(0 \leq x \leq 1\). Prove that if \(f\) is continuous on \([0,1]\) there is at least one number \(c\) in \([0,1]\) such that \(f(c)=c\). (HINT: If neither 0 nor 1 qualifies as \(c\), then \(f(0)>0\) and \(f(1)<1 .\) Consider the function \(g\) for which \(g(x)=f(x)-x\) and apply the intermediate-value theorem to \(g\) on \([0,1]\).)

\(\sum_{k=1}^{100}\left[\frac{1}{k}-\frac{1}{k+1}\right]\)

Express as a definite integral: \(\lim _{n\lrcorner+\infty} \sum_{i=1}^{n}\left(8 i^{2} / n^{3}\right)\). (HINT: Consider the function \(f\) for which \(f(x)=x^{2}\).)

\(D_{x} \int_{-x}^{x} \frac{d t}{3+t^{2}}\)