91影视

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

$\frac{d}{dx}{鈭�}_0^x鈥�e^{鈭�t^2}dt$

The area between the graphs of$f\left(x\right)=4-x^{2}and1-2xon[-4,4].$

Calculate areas and average values : Express each of the following areas or average values with a definite integral, and then use definite integral formulas to compute the exact value of the area or average value.

The area between the graphs of $f\left(x\right)=4-x^2.$and $1-2x$on $\left[-4,4\right].$

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.

$f\left(x\right)=\sin 鈦�x,g\left(x\right)=\cos 鈦�x,[0,蟺]$

For each function f and interval [a, b] in Exercises 34鈥�38, it is possible to find the exact signed area between the graph of f and the x-axis on [a, b] geometrically by using the areas of circles,

triangles, and rectangles. Find this exact area, and then calculate the left, right, midpoint, upper, lower, and trapezoid sums with n = 4. Which approximation rule is most accurate?

$f\left(x\right)=3+\sqrt{4-x^2},\left[a,b\right]=\left[-2,2\right]$

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.

$鈭�\frac{x^{-1}}{1-x^{-1}}dx$

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

${鈭�}_0^{蟺}鈥�\left({\sin }^2鈦�x+{\cos }^2鈦�x\right)dx$

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.

${鈭�}_{-2}^6\left(4f\right(x)-2)dx$.

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

$鈭�cs{c}^{2}xdx$$=.........$

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.

$f\left(x\right)=\sqrt{x-1},\left[a,b\right]=\left[2,3\right],rightsum,n=4$