91Ó°ÊÓ

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

$∫\sin hxdx=................$

Combining derivatives and integrals: Simplify each of the following as much as possible:

$\frac{d}{dx}{∫}_0^x{e}^{-t^2}dt$

In Exercises 39–44, write out the sigma notation for the Riemann sum described in such a way that the only letter which appears in the general term of the sum is k. Don’t calculate the value of the sum; just write it down in sigma notation.

$f\left(x\right)=\sqrt{1-x^2},\left[a,b\right]=\left[-1,1\right],midpointsum,n=20$

For each definite integral in Exercises 41–46, (a) find the general n-rectangle right sum and simplify your answer with sum formulas. Then (b) use your answer to approximate the definite integral with $n=100$and $n=1000$. Finally, (c) take the limit as $n→∞$to find the exact value.

${∫}_{-3}^2{x}^{2}dx$

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.

$∫(x^2{e}^x+2x{e}^x)dx$

Combining derivatives and integrals: Simplify each of the following as much as possible:

$\frac{d}{dx}{∫}_0^{\ln x}{\sin }^{3}tdt$

Use the Fundamental Theorem of Calculus to find the exact

values of each of the definite integrals in Exercises 19–64. Use

a graph to check your answer. (Hint: The integrands that involve

absolute values will have to be considered piecewise.)

${∫}_0^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{1}{\sqrt{9-x^2}}dx$

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 $x=aandx=b$.

$f\left(x\right)=x^2,g\left(x\right)=x+2,[-3,3]$

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

$∫\cosh xdx=........$

Suppose that, as in Section 4.1, you drive in a car for 40 seconds with velocity $v\left(t\right)=-0.22t^2+88t$feet per second second, as shown in the graph that follows. If your total

distance travelled is equal to the area under the velocity curve on [0, 40], then find lower and upper bounds for your distance travelled by using

(a) the lower sum with n = 4 rectangles;

(b) the upper sum with n = 4 rectangles.