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.

$f\left(x\right)=\ln \left(x\right),\left[a,b\right]=\left[2,5\right],leftsum,n=100$

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

${鈭�}_{鈭�3}^2鈥�\frac{x^2+5}{2x}dx$

Write each expression in Exercises 41鈥�43 in one sigma notation (with some extra terms added to or subtracted from the sum, as necessary).

$\underset{k=1}{\overset{40}{鈭�}}\frac{1}{k}-\underset{k=0}{\overset{39}{鈭�}}\frac{1}{k+1}$

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

$\frac{d}{dx}{鈭�}_1^x鈥�\ln 鈦�tdt$

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.

$鈭�\frac{1}{4+x^2}dx$

For each pair of functions $fandg$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$x=atox=b.$

$f\left(x\right)=1+x,g\left(x\right)=x+2,[0,3]$

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.

$鈭�\frac{2x}{1+x^2}dx$

Write each expression in Exercises 41鈥�43 in one sigma notation (with some extra terms added to or subtracted from the sum, as necessary).

$3\underset{k=2}{\overset{25}{鈭�}}{k}^2+2\underset{k=2}{\overset{24}{鈭�k}}-\underset{k=0}{\overset{25}{鈭�}}1$

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

$\frac{d^2}{d{x}^2}{鈭�}_{x^2}^1鈥�\ln 鈦�\left|t\right|dt$

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.

$f\left(x\right)=\sin x,\left[a,b\right]=\left[0,\mathrm{蟺}\right],trapezoidsum,n=4$