Chapter 4

In Problems \(1-14\), find the average value of the function on the given interval. $$ f(x)=4 x^{3} ; \quad[1,3] $$

Evaluate the indicated integrals. $$ \int_{0}^{1}\left(x^{3}-3 x^{2}+3 \sqrt{x}\right) d x $$

In Problems \(1-8\), find the value of the indicated sum. \(\sum_{k=1}^{6}(k-1)\)

Use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{0}^{2} x^{3} d x $$

In Problems \(1-6,\) use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{1}^{3} \frac{1}{x^{2}} d x $$

Find the average value of the function on the given interval. $$ f(x)=5 x^{2} ; \quad[1,4] $$

Evaluate the indicated integrals. $$ \int_{1}^{2} \frac{2 x^{4}-3 x^{2}+1}{x^{2}} d x $$

Find the value of the indicated sum. \(\sum_{i=1}^{6} i^{2}\)

Use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{-1}^{2} x^{4} d x $$

In Problems \(1-6,\) use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{1}^{3} \frac{1}{x^{3}} d x $$

Evaluate the indicated integrals. $$ \int_{1}^{\pi} \frac{y^{3}-9 y \sin y+26 y^{-1}}{y} d y $$

Find the value of the indicated sum. \(\sum_{k=1}^{7} \frac{1}{k+1}\)

Use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{-1}^{2}\left(3 x^{2}-2 x+3\right) d x $$

In Problems \(1-6,\) use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{0}^{2} \sqrt{x} d x $$

Find the average value of the function on the given interval. $$ f(x)=\frac{x^{2}}{\sqrt{x^{3}+16}} ; \quad[0,2] $$

Evaluate the indicated integrals. $$ \int_{4}^{9} y \sqrt{y^{2}-4} d y $$

Find the value of the indicated sum. \(\sum_{l=3}^{8}(l+1)^{2}\)

Use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{1}^{2}\left(4 x^{3}+7\right) d x $$

In Problems \(1-6,\) use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{1}^{3} x \sqrt{x^{2}+1} d x $$

Calculate the Riemann sum \(\sum_{i=1}^{n} f\left(\bar{x}_{i}\right) \Delta x_{i}\) for the given data. \(f(x)=x^{2} / 2+x ;[-2,2]\) is divided into eight equal subintervals, \(\bar{x}_{i}\) is the midpoint.

Find the average value of the function on the given interval. $$ f(x)=2+|x| ; \quad[-2,1] $$

Evaluate the indicated integrals. $$ \int_{2}^{8} z\left(2 z^{2}-3\right)^{1 / 3} d z $$

Find the value of the indicated sum. \(\sum_{m=1}^{8}(-1)^{m} 2^{m-2}\)

Use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{1}^{4} \frac{1}{w^{2}} d w $$

In Problems \(1-6,\) use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{0}^{1} x\left(x^{2}+1\right)^{5} d x $$

Calculate the Riemann sum \(\sum_{i=1}^{n} f\left(\bar{x}_{i}\right) \Delta x_{i}\) for the given data. \(f(x)=4 x^{3}+1 ;[0,3]\) is divided into six equal subintervals, \(\bar{x}_{i}\) is the right end point.

Find the average value of the function on the given interval. $$ f(x)=x+|x| ; \quad[-3,2] $$

Evaluate the indicated integrals. $$ \int_{0}^{\pi / 2} \cos ^{4} x \sin x d x $$

Find the value of the indicated sum. \(\sum_{k=3}^{7} \frac{(-1)^{k} 2^{k}}{(k+1)}\)

Use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{1}^{3} \frac{2}{t^{3}} d t $$

In Problems \(1-6,\) use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{1}^{4}(x+1)^{3 / 2} d x $$

Find the average value of the function on the given interval. $$ f(x)=\cos x ; \quad[0, \pi] $$

Evaluate the indicated integrals. $$ \int_{0}^{\pi}(x+1) \tan ^{2}\left(3 x^{2}+6 x\right) \sec ^{2}\left(3 x^{2}+6 x\right) d x $$

Find the value of the indicated sum. \(\sum_{n=1}^{6} n \cos (n \pi)\)

Use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{0}^{4} \sqrt{t} d t $$

Use the given values of \(a\) and \(b\) and express the given limit as a definite integral. $$ \lim _{|P| \rightarrow 0} \sum_{i=1}^{n}\left(\bar{x}_{i}+1\right)^{3} \Delta x_{i} ; a=0, b=2 $$

Find the average value of the function on the given interval. $$ f(x)=\sin x ;[0, \pi] $$

Evaluate the indicated integrals. $$ \int_{0}^{2} \frac{t^{3}}{\sqrt{t^{4}+9}} d t $$

Find the value of the indicated sum. \(\sum_{k=-1}^{6} k \sin (k \pi / 2)\)

Use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{1}^{8} \sqrt[3]{w} d w $$

In Problems \(7-10,\) use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) midpoint Riemann sum, (4) Trapezoidal Rule, (5) Parabolic Rule with \(n=4,8,16\). (Note that none of these can be evaluated using the Second Fundamental Theorem of Calculus with the techniques you have learned so far.) Present your approximations in a table like this: $$ \int_{1}^{3} \frac{1}{x} d x $$

Use the given values of \(a\) and \(b\) and express the given limit as a definite integral. $$ \lim _{|P| \rightarrow 0} \sum_{i=1}^{n} \frac{\bar{x}_{i}^{2}}{1+\bar{x}_{i}} \Delta x_{i} ; a=-1, b=1 $$

Find the average value of the function on the given interval. $$ f(x)=x \cos x^{2} ; \quad[0, \sqrt{\pi}] $$

Evaluate the indicated integrals. $$ \int_{1}^{2} t^{4}\left(t^{5}+5\right)^{2 / 3} d t $$

Suppose that \(\int_{0}^{1} f(x) d x=2, \int_{1}^{2} f(x) d x=3, \int_{0}^{1} g(x) d x=-1\) and \(\int_{0}^{2} g(x) d x=4\). Use properties of definite integrals (linearity, interval additivity, and so on) to calculate each of the integrals. $$ \int_{1}^{2} 2 f(x) d x $$

In Problems 9-14, write the indicated sum in sigma notation. \(1+2+3+\cdots+41\)

Use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{-4}^{-2}\left(y^{2}+\frac{1}{y^{3}}\right) d y $$

In Problems \(7-10,\) use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) midpoint Riemann sum, (4) Trapezoidal Rule, (5) Parabolic Rule with \(n=4,8,16\). (Note that none of these can be evaluated using the Second Fundamental Theorem of Calculus with the techniques you have learned so far.) Present your approximations in a table like this: $$ \int_{0}^{2} \sqrt{x^{2}+1} d x $$

Use the given values of \(a\) and \(b\) and express the given limit as a definite integral. $$ \lim _{|P| \rightarrow 0} \sum_{i=1}^{n}\left(\sin \bar{x}_{i}\right)^{2} \Delta x_{i} ; a=0, b=\pi $$

Find the average value of the function on the given interval. $$ f(x)=\sin ^{2} x \cos x ; \quad[0, \pi / 2] $$