91影视

For each definite integral in Exercises 41鈥�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 $n鈫�鈭�$to find the exact value.

${鈭�}_0^{1}2x^{2}dx$

For each pair of functions f and g and interval$[a,b]$ in Exercises 41鈥�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 $x=aandx=b.$

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

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

$鈭�\frac{1}{\left|x\right|\sqrt{x^2-1}}dx=...............$

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${鈭�}_1^4鈥�\frac{2x+3}{x^2+3x+4}dx$

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

$\frac{d}{dx}{鈭�}_0^3{e}^{-t^2}dt$

For each pair of functions $f$and $g$and interval $[a,b]$in Exercises 41鈥�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 $x=aandx=b.$.

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

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.

$\frac{d^2}{d{x}^2}{鈭�}_1^{e^x}鈥�f\left(t\right)g\left(t\right)dt$

Use integration formulas to solve each integral in Exercises 21鈥�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.

$鈭�2\sin x\cos xdx$

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${鈭�}_{蟺/4}^{蟺/2}鈥�\csc 鈦�x\cot 鈦�xdx$

Find the sum or quantity without completely expanding or calculating any sums.

Given$\underset{k=3}{\overset{10}{鈭�}}{a}_k=12$and$\underset{k=2}{\overset{10}{鈭�}}{a}_k=23$, find$a_2$.