Chapter 4

Use summation rules to compute the sum. $$\sum_{i=1}^{200}\left(4-3 i-i^{2}\right)$$

Find the general antiderivative. $$\int 5 \sec ^{2} x d x$$

Evaluate the indicated integral. $$\int \frac{\cos (1 / x)}{x^{2}} d x$$

Evaluate the derivative using properties of logarithms where needed. $$\frac{d}{d x}(\ln \sqrt{\frac{x^{3}}{x^{5}+1}})$$

Compute the exact value and compute the error (the difference between the approximation and the exact value) in each of the Midpoint, Trapezoidal and Simpson's Rule approximations using \(n=10, n=20, n=40\) and \(n=80\). $$\int_{0}^{\pi / 4} \cos x d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{1}^{2} \frac{x^{2}-3 x+4}{x^{2}} d x$$

Write the given (total) area as an integral or sum of integrals. The area between \(y=\sin x \quad\) and the \(\quad x\) -axis for \(-\pi / 2 \leq x \leq \pi / 4\).

Use Riemann sums and a limit to compute the exact area under the curve. $$y=x^{2}+3 x \text { on }[0,1]$$

Find the general antiderivative. $$\int 4 \frac{\cos x}{\sin ^{2} x} d x$$

Use summation rules to compute the sum. $$\sum_{i=1}^{250}\left(i^{2}+8\right)$$

Evaluate the indicated integral. $$\int \frac{1}{\sqrt{x}(\sqrt{x}+1)^{2}} d x$$

Evaluate the integral. $$\int \frac{3 x^{3}}{x^{4}+5} d x$$

Fill in the blanks with the most appropriate power of 2(2,4,8 etc.). If you double \(n,\) the error in the Midpoint Rule is divided by____ \(\quad\) If you double \(n,\) the error in the Trapezoidal Rule is divided by_____ \(\quad\) If you double \(n,\) the error in Simpson's Rule is divided by_____.

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{4} x(x-2) d x$$

Write the given (total) area as an integral or sum of integrals. The area between \(y=x^{3}-3 x^{2}+2 x\) and the \(x\) -axis for \(0 \leq x \leq 2\).

Use Riemann sums and a limit to compute the exact area under the curve. $$y=2 x^{2}+1 \text { on }[1,3]$$

Use summation rules to compute the sum. $$\sum_{i=3}^{n}\left(i^{2}-3\right)$$

Find the general antiderivative. $$\int\left(3 e^{x}-2\right) d x$$

Evaluate the indicated integral. $$\int \frac{x}{x^{2}+4} d x$$

Evaluate the integral. $$\int \frac{1}{\sqrt{x}(\sqrt{x}+1)} d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{\pi / 3} \frac{3}{\cos ^{2} x} d x$$

Write the given (total) area as an integral or sum of integrals. The area between \(y=x^{3}-4 x\) and the \(x\) -axis for \(-2 \leq x \leq 3\).

Use Riemann sums and a limit to compute the exact area under the curve. $$y=4 x+2 \text { on }[1,3]$$

Find the general antiderivative. $$\int\left(4 x-2 e^{x}\right) d x$$

Use summation rules to compute the sum. $$\sum_{i=0}^{n}\left(i^{2}+5\right)$$

Evaluate the indicated integral. $$\int \frac{4}{x(\ln x+1)^{2}} d x$$

Approximate the given value using (a) Midpoint Rule, (b) Trapezoidal Rule and (c) Simpson's Rule with \(n=4\). $$\ln 4=\int_{1}^{4} \frac{1}{x} d x$$

Evaluate the integral. $$\int \frac{1}{x \ln x} d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{\ln 2}\left(e^{x / 2}\right)^{2} d x$$

Use the given velocity function and initial position to estimate the final position \(s(b)\). $$v(t)=40\left(1-e^{-2 t}\right), s(0)=0, b=4$$

Use the given function values to estimate the area under the curve using left- endpoint and right-endpoint evaluation. $$\begin{array}{|l|r|r|r|r|r|r|r|r|r|} \hline x & 0.0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 \\ \hline f(x) & 2.0 & 2.4 & 2.6 & 2.7 & 2.6 & 2.4 & 2.0 & 1.4 & 0.6 \\ \hline \end{array}$$

Find the general antiderivative. $$\int(3 \cos x-1 / x) d x$$

Compute the sum and the limit of the sum as \(n \rightarrow \infty.\) $$\sum_{i=1}^{n} \frac{1}{n}\left[\left(\frac{i}{n}\right)^{2}+2\left(\frac{i}{n}\right)\right]$$

Evaluate the indicated integral. $$\int \tan 2 x \, d x$$

Approximate the given value using (a) Midpoint Rule, (b) Trapezoidal Rule and (c) Simpson's Rule with \(n=4\). $$\ln 8=\int_{1}^{8} \frac{1}{x} d x$$

Evaluate the integral. $$\int \frac{1}{\sqrt{1-x^{2}} \sin ^{-1} x} d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{\pi}\left(\sin ^{2} x+\cos ^{2} x\right) d x$$

Use the given velocity function and initial position to estimate the final position \(s(b)\). $$v(t)=30 e^{-l / 4}, s(0)=-1, b=4$$

Use the given function values to estimate the area under the curve using left- endpoint and right-endpoint evaluation. $$\begin{array}{|l|r|r|r|r|r|r|r|r|r|} \hline x & 0.0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 & 1.6 \\ \hline f(x) & 2.0 & 2.2 & 1.6 & 1.4 & 1.6 & 2.0 & 2.2 & 2.4 & 2.0 \\ \hline \end{array}$$

Find the general antiderivative. $$\int\left(2 x^{-1}+\sin x\right) d x$$

Compute the sum and the limit of the sum as \(n \rightarrow \infty.\) $$\sum_{i=1}^{n} \frac{1}{n}\left[\left(\frac{i}{n}\right)^{2}-5\left(\frac{i}{n}\right)\right]$$

Evaluate the indicated integral. $$\int \frac{\left(\sin ^{-1} x\right)^{3}}{\sqrt{1-x^{2}}} d x$$

Approximate the given value using (a) Midpoint Rule, (b) Trapezoidal Rule and (c) Simpson's Rule with \(n=4\). $$\sin 1=\int_{0}^{1} \cos x d x$$

Evaluate the integral. $$\int \frac{e^{2 x}}{1+e^{2 x}} d x$$

Use the Fundamental Theorem if possible or estimate the integral using Riemann sums. $$\int_{0}^{2} \sqrt{x^{2}+1} d x$$

Use Theorem 4.2 to write the expression as a single integral. $$\int_{0}^{2} f(x) d x+\int_{2}^{3} f(x) d x$$

Use the given function values to estimate the area under the curve using left- endpoint and right-endpoint evaluation. $$\begin{array}{|l|r|r|r|r|r|r|r|r|r|} \hline x & 1.0 & 1.1 & 1.2 & 1.3 & 1.4 & 1.5 & 1.6 & 1.7 & 1.8 \\ \hline f(x) & 1.8 & 1.4 & 1.1 & 0.7 & 1.2 & 1.4 & 1.8 & 2.4 & 2.6 \\ \hline \end{array}$$

Find the general antiderivative. $$\int \frac{4 x}{x^{2}+4} d x$$

Compute the sum and the limit of the sum as \(n \rightarrow \infty.\) $$\sum_{i=1}^{n} \frac{1}{n}\left[4\left(\frac{2 i}{n}\right)^{2}-\left(\frac{2 i}{n}\right)\right]$$

Evaluate the indicated integral. $$\int x^{2} \sec ^{2} x^{3} d x$$