91Ó°ÊÓ

For each pair of functions $f$and $g$and interval $[a,b]$in Exercises 41–52, use definite integrals and the Fundamental Theorem of Calculus to find the exact area of the region between the graphs of f and g from $x=atox=b.$

$f\left(x\right)=x^3,g\left(x\right)=(x-2{)}^3,[a,b]=[-1,2]$

Find a function $f$that has the given derivative $f\text{'}$and value $f\left(c\right)$. Find an antiderivative of $f\text{'}$by hand, if possible; if it is not possible to antidifferentiation by hand, use the Second Fundamental Theorem of Calculus to write down an antiderivative.

$f^{\mathrm{′}}\left(x\right)=\frac{1}{2x−1},f\left(1\right)=3$

Calculate the exact value of each definite integral in Exercises 47–52 by using properties of definite integrals and the formulas in Theorem 4.13.

${∫}_5^2\left(9+10x-x^2\right)dx$

$$\begin{gathered} \mathrm{To}\mathrm{approximate}\mathrm{the}\mathrm{flow}\mathrm{f}\left(\mathrm{t}\right)\mathrm{of}\mathrm{the}\mathrm{Lochsa}\mathrm{River}\mathrm{in}\mathrm{its}\mathrm{flood}\mathrm{stage},\mathrm{we}\mathrm{can}\mathrm{use}\mathrm{a}\mathrm{function}\mathrm{of}\mathrm{the}\mathrm{form} \\ \mathrm{g}\left(\mathrm{t}\right)={\mathrm{c}}_1+{\mathrm{c}}_2\left(\sin \left(\frac{(\mathrm{t}-90)\mathrm{π}}{105}\right)-\frac{2}{\mathrm{π}}\right), \\ \mathrm{where}\mathrm{the}\mathrm{coefficients}\mathrm{c}1\mathrm{and}\mathrm{c}2\mathrm{are}\mathrm{found}\mathrm{by}\mathrm{evaluating}\mathrm{the}\mathrm{following}\mathrm{two}\mathrm{integrals}: \\ {\mathrm{c}}_1=\frac{1}{105}{∫}_{90}^{195}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}, \\ {\mathrm{c}}_2=\frac{1}{9.95}{∫}_{90}^{195}\mathrm{f}\left(\mathrm{t}\right)\left(\sin \left(\frac{(\mathrm{t}-90)\mathrm{π}}{105}\right)-\frac{2}{\mathrm{π}}\right)\mathrm{dt}. \\ \left(\mathrm{a}\right)\mathrm{Use}\mathrm{the}\mathrm{data}\mathrm{points}(\mathrm{t},\mathrm{f}(\mathrm{t}\left)\right)=(90,2100),(120,6300),(150,11000),\mathrm{and}(180,4000),\mathrm{and} \\ \mathrm{left}\mathrm{Riemann}\mathrm{sums}\mathrm{to}\mathrm{approximate}\mathrm{the}\mathrm{values}\mathrm{of}\mathrm{the}\mathrm{integrals}\mathrm{for}\mathrm{c}1\mathrm{and}\mathrm{c}2. \\ \left(\mathrm{b}\right)\mathrm{Now}\mathrm{that}\mathrm{you}\mathrm{have}\mathrm{found}\mathrm{c}1\mathrm{and}\mathrm{c}2,\mathrm{plot}\mathrm{the}\mathrm{resulting}\mathrm{function}\mathrm{g}\left(\mathrm{t}\right)\mathrm{against}\mathrm{the}\mathrm{data}\mathrm{points}\mathrm{from}\mathrm{Exercise}48. \end{gathered}$$

Determine which of the limit of sums in Exercises 47–52 are infinite and which are finite. For each limit of sums that is finite, compute its value

$\underset{n→∞}{\lim }\underset{k=1}{\overset{n}{∑}}\frac{{\left(k+1\right)}^2}{n^3-1}$

Integral Formulas: Fill in the blanks to complete each of the following integration formulas.

$∫\frac{1}{1-x^2}dx=...$.

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: $\tan \left(x\right)=\frac{\sin \left(x\right)}{\cos \left(x\right)}$).

$∫\frac{e^{3x}-2e^{4x}}{e^{2x}}dx$.

Find the derivative and an antiderivative of each of the following functions:

$$\begin{gathered} f\left(x\right)={\mathrm{Ï€}}^2 \\ f\left(x\right)={\left(\sqrt[11]{x}\right)}^{-2} \\ f\left(x\right)=\frac{1}{5x} \\ f\left(x\right)=se{c}^2(3x+1) \end{gathered}$$

Approximating the length of a curve: Suppose you want to

calculate the driving distance between New York City and

Dallas, Texas.
How accurate do you think your approximation is? Is it an over-approximation or an under-approximation?

Functions defined by area accumulation: We know that for fixed real numbers a and b and an integrable functionf, the definite integral ${∫}_a^{b}f\left(x\right)dx$is a real number. For different real values of b, we get (potentially) different values for the integral ${∫}_a^{b}f\left(x\right)dx.$

What is the relationship between the formula that describes ${∫}_0^{b}2xdx$and the formula that describes${∫}_1^{b}2xdx$