Section 5.6 : Definition of the Definite Integral

For problems 1 – 4 use the definition of the definite integral to evaluate the integral. Use the right end point of each interval for \(x_{\,i}^*\).

1. \( \displaystyle \int_{{ - 2}}^{1}{{7 - 4x\,dx}}\)
2. \( \displaystyle \int_{0}^{2}{{3{x^2} + 4x\,dx}}\)
3. \( \displaystyle \int_{{ - 1}}^{1}{{{{\left( {x - 3} \right)}^2}\,dx}}\)
4. \( \displaystyle \int_{0}^{3}{{8{x^3} + 3x - 2\,dx}}\)
5. Evaluate : \( \displaystyle \int_{{ - 123}}^{{ - 123}}{{{{\cos }^6}\left( {2x} \right) - {{\sin }^8}\left( {4x} \right)\,dx}}\)

For problems 6 – 8 determine the value of the given integral given that \( \displaystyle \int_{{ - 2}}^{5}{{f\left( x \right)\,dx}} = 1\) and \( \displaystyle \int_{{ - 2}}^{5}{{g\left( x \right)\,dx}} = 8\).

1. \( \displaystyle \int_{{ - 2}}^{5}{{ - 3g\left( x \right)\,dx}}\)
2. \( \displaystyle \int_{{ - 2}}^{5}{{7f\left( x \right) - \frac{1}{4}g\left( x \right)\,dx}}\)
3. \( \displaystyle \int_{5}^{{ - 2}}{{12g\left( x \right) - 3f\left( x \right)\,dx}}\)
4. Determine the value of \( \displaystyle \int_{7}^{{ - 1}}{{f\left( x \right)\,dx}}\) given that \( \displaystyle \int_{{13}}^{7}{{f\left( x \right)\,dx}} = - 9\) and \( \displaystyle \int_{{13}}^{{ - 1}}{{f\left( x \right)\,dx}} = - 12\).
5. Determine the value of \( \displaystyle \int_{0}^{6}{{4f\left( x \right)\,dx}}\) given that \( \displaystyle \int_{0}^{5}{{f\left( x \right)\,dx}} = 10\) and \( \displaystyle \int_{5}^{6}{{f\left( x \right)\,dx}} = 3\).
6. Determine the value of \( \displaystyle \int_{2}^{{10}}{{f\left( x \right)\,dx}}\) given that \( \displaystyle \int_{2}^{4}{{f\left( x \right)\,dx}} = - 1\), \( \displaystyle \int_{4}^{7}{{f\left( x \right)\,dx}} = 3\) and \( \displaystyle \int_{{10}}^{7}{{f\left( x \right)\,dx}} = - 8\).
7. Determine the value of \( \displaystyle \int_{{ - 5}}^{{ - 1}}{{f\left( x \right)\,dx}}\) given that \( \displaystyle \int_{2}^{{ - 5}}{{f\left( x \right)\,dx}} = 56\), \( \displaystyle \int_{7}^{2}{{f\left( x \right)\,dx}} = - 90\) and \( \displaystyle \int_{{ - 1}}^{7}{{f\left( x \right)\,dx}} = 45\).

For problems 13 – 17 sketch the graph of the integrand and use the area interpretation of the definite integral to determine the value of the integral.

1. \( \displaystyle \int_{{ - 2}}^{1}{{12 - 5x\,dx}}\)
2. \( \displaystyle \int_{0}^{4}{{\sqrt {16 - {x^2}} \,dx}}\)
3. \( \displaystyle \int_{{ - 3}}^{3}{{5 - \sqrt {9 - {x^2}} \,dx}}\)
4. \( \displaystyle \int_{{ - 1}}^{3}{{8x - 3\,dx}}\)
5. \( \displaystyle \int_{1}^{6}{{\left| {x - 3} \right|\,dx}}\)

For problems 18 – 23 differentiate each of the following integrals with respect to x.

1. \( \displaystyle \int_{{ - 8}}^{x}{{{{\bf{e}}^{\cos \left( t \right)}}\,dt}}\)
2. \( \displaystyle \int_{2}^{{{x^{\,2}}}}{{\sqrt {\cos \left( t \right) + 3} \,dt}}\)
3. \( \displaystyle \int_{0}^{{{{\bf{e}}^{3x}}}}{{\frac{1}{{{t^4} + {t^2} + 1}}dt}}\)
4. \( \displaystyle \int_{{\sin \left( {9x} \right)}}^{8}{{\frac{{{{\bf{e}}^t}}}{{7t}}dt}}\)
5. \( \displaystyle \int_{{{x^{\,3}}}}^{x}{{{{\cos }^4}\left( t \right) - {{\sin }^2}\left( t \right)\,dt}}\)
6. \( \displaystyle \int_{{9x}}^{{\tan \left( x \right)}}{{\frac{{\cos \left( t \right) + 2}}{{\sin \left( t \right) + 4}}\,dt}}\)
7. Evaluate the limit : \(\mathop {\lim }\limits_{x \to 0} \frac{{\displaystyle \int_{0}^{x}{{{{\bf{e}}^{{t^2}}}\,dt}}}}{x}\)