91影视

In Exercises 39鈥�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鈥檛 calculate the value of the sum; just write it down in sigma notation.

$f\left(x\right)=x^2,\left[a,b\right]=\left[0,3\right],leftsum,n=3$

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

$鈭�cscxcotxdx=.............$

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.

$鈭�\frac{-3}{1+16x^2}dx.$

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals.

Use a graph to check your answer.

${鈭�}_{鈭�3}^2鈥�\frac{1}{(x+5{)}^2}dx$

For each pair of functions f and g and each interval [a, b] in Exercises 38鈥�40, use at least eight rectangles to approximate the area of the region between the graphs of f and g on the interval [a, b]. Your work should include graphs of f and g together with the rectangles that you used.

$f\left(x\right)=e^x,g\left(x\right)=\ln 鈦�x,[0.5,2.5]$

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

$\frac{d}{dx}{鈭�}_0^x鈥�t^{3}dt$

Find a formula for each of the sums in Exercises 40, and then use these formulas to calculate each sum for $n=100,500$and$1000$

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

$\frac{d}{dx}\left({\left({鈭�}_{\sin 鈦�x}^{蟺}鈥�\frac{t}{\cos 鈦�t}dt\right)}^3\right)$

If ${鈭�}_{-2}^{3}f\left(x\right)dx=4,{鈭�}_{-2}^{6}f\left(x\right)dx=9,{鈭�}_{-2}^{3}g\left(x\right)dx=2$, and ${鈭�}_3^{6}g\left(x\right)dx=3$, then find the values of each definite integral in Exercises $29-40$. If there is not enough information, explain why.

${鈭�}_{-2}^{-2}x\left(f\right(x)+3{)}^{2}dx$

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.

${鈭�}_2^5(5-x)dx$