91Ó°ÊÓ

Evaluating a Sum In Exercises \(21-24,\) use the summation formulas to rewrite the expression without the summation notation. Use the result to find the sums for \(n=10,100,1000,\) and \(10,000\) . $$ \sum_{i=1}^{n} \frac{2 i+1}{n^{2}} $$

Evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{-1}^{0}\left(t^{1 / 3}-t^{2 / 3}\right) d t\)

Evaluating a Sum In Exercises \(21-24,\) use the summation formulas to rewrite the expression without the summation notation. Use the result to find the sums for \(n=10,100,1000,\) and \(10,000\) . $$ \sum_{j=1}^{n} \frac{7 j+4}{n^{2}} $$

Evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{-8}^{-1} \frac{x-x^{2}}{2 \sqrt[3]{x}} d x\)

Finding an Indefinite Integral In Exercises \(5-26\) , find the indefinite integral and check the result by differentiation. $$ \int \frac{x^{3}}{\sqrt{1+x^{4}}} d x $$

In Exercises 11–32, find the indefinite integral and check the result by differentiation. $$ \int \frac{x^{4}-3 x^{2}+5}{x^{4}} d x $$

In Exercises 11–32, find the indefinite integral and check the result by differentiation. $$ \int(x+1)(3 x-2) d x $$

Finding an Indefinite Integral In Exercises \(5-26\) , find the indefinite integral and check the result by differentiation. $$ \int\left(1+\frac{1}{t}\right)^{3}\left(\frac{1}{t^{2}}\right) d t $$

Evaluating a Definite Integral Using a Geometric Formula In Exercises \(23-32,\) sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral \((a>0, r>0) .\) $$ \int_{0}^{3} 4 d x $$

Evaluating a Sum In Exercises \(21-24,\) use the summation formulas to rewrite the expression without the summation notation. Use the result to find the sums for \(n=10,100,1000,\) and \(10,000\) . $$ \sum_{k=1}^{n} \frac{6 k(k-1)}{n^{3}} $$