I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
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 \(x_{\,i}^*\).
- \( \displaystyle \int_{{ - 2}}^{1}{{7 - 4x\,dx}}\)
- \( \displaystyle \int_{0}^{2}{{3{x^2} + 4x\,dx}}\)
- \( \displaystyle \int_{{ - 1}}^{1}{{{{\left( {x - 3} \right)}^2}\,dx}}\)
- \( \displaystyle \int_{0}^{3}{{8{x^3} + 3x - 2\,dx}}\)
- 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 \( \displaystyle \int_{{ - 2}}^{5}{{g\left( x \right)\,dx}} = 8\).
- \( \displaystyle \int_{{ - 2}}^{5}{{ - 3g\left( x \right)\,dx}}\)
- \( \displaystyle \int_{{ - 2}}^{5}{{7f\left( x \right) - \frac{1}{4}g\left( x \right)\,dx}}\)
- \( \displaystyle \int_{5}^{{ - 2}}{{12g\left( x \right) - 3f\left( x \right)\,dx}}\)
- 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 \( \displaystyle \int_{{13}}^{{ - 1}}{{f\left( x \right)\,dx}} = - 12\).
- 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 \( \displaystyle \int_{5}^{6}{{f\left( x \right)\,dx}} = 3\).
- 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 \( \displaystyle \int_{{10}}^{7}{{f\left( x \right)\,dx}} = - 8\).
- 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.
- \( \displaystyle \int_{{ - 2}}^{1}{{12 - 5x\,dx}}\)
- \( \displaystyle \int_{0}^{4}{{\sqrt {16 - {x^2}} \,dx}}\)
- \( \displaystyle \int_{{ - 3}}^{3}{{5 - \sqrt {9 - {x^2}} \,dx}}\)
- \( \displaystyle \int_{{ - 1}}^{3}{{8x - 3\,dx}}\)
- \( \displaystyle \int_{1}^{6}{{\left| {x - 3} \right|\,dx}}\)
For problems 18 – 23 differentiate each of the following integrals with respect to x.
- \( \displaystyle \int_{{ - 8}}^{x}{{{{\bf{e}}^{\cos \left( t \right)}}\,dt}}\)
- \( \displaystyle \int_{2}^{{{x^{\,2}}}}{{\sqrt {\cos \left( t \right) + 3} \,dt}}\)
- \( \displaystyle \int_{0}^{{{{\bf{e}}^{3x}}}}{{\frac{1}{{{t^4} + {t^2} + 1}}dt}}\)
- \( \displaystyle \int_{{\sin \left( {9x} \right)}}^{8}{{\frac{{{{\bf{e}}^t}}}{{7t}}dt}}\)
- \( \displaystyle \int_{{{x^{\,3}}}}^{x}{{{{\cos }^4}\left( t \right) - {{\sin }^2}\left( t \right)\,dt}}\)
- \( \displaystyle \int_{{9x}}^{{\tan \left( x \right)}}{{\frac{{\cos \left( t \right) + 2}}{{\sin \left( t \right) + 4}}\,dt}}\)
- Evaluate the limit : \(\mathop {\lim }\limits_{x \to 0} \frac{{\displaystyle \int_{0}^{x}{{{{\bf{e}}^{{t^2}}}\,dt}}}}{x}\)