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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 : $$\mathop {\lim }\limits_{x \to 0} \frac{{\displaystyle \int_{0}^{x}{{{{\bf{e}}^{{t^2}}}\,dt}}}}{x}$$