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### Section 5.6 : Definition of the Definite Integral

For problems 1 & 2 use the definition of the definite integral to evaluate the integral. Use the right end point of each interval for $$x_{\,i}^*$$.

1. $$\displaystyle \int_{1}^{4}{{2x + 3\,dx}}$$ Solution
2. $$\displaystyle \int_{0}^{1}{{6x\left( {x - 1} \right)\,dx}}$$ Solution
3. Evaluate : $$\displaystyle \int_{4}^{4}{{\frac{{\cos \left( {{{\bf{e}}^{3x}} + {x^2}} \right)}}{{{x^4} + 1}}\,dx}}$$ Solution

For problems 4 & 5 determine the value of the given integral given that $$\displaystyle \int_{6}^{{11}}{{f\left( x \right)\,dx}} = - 7$$ and $$\displaystyle \int_{6}^{{11}}{{g\left( x \right)\,dx}} = 24$$.

1. $$\displaystyle \int_{{11}}^{6}{{9f\left( x \right)\,dx}}$$ Solution
2. $$\displaystyle \int_{6}^{{11}}{{6g\left( x \right) - 10f\left( x \right)\,dx}}$$ Solution
3. Determine the value of $$\displaystyle \int_{2}^{9}{{f\left( x \right)\,dx}}$$ given that $$\displaystyle \int_{5}^{2}{{f\left( x \right)\,dx}} = 3$$ and $$\displaystyle \int_{5}^{9}{{f\left( x \right)\,dx}} = 8$$. Solution
4. Determine the value of $$\displaystyle \int_{{ - 4}}^{{20}}{{f\left( x \right)\,dx}}$$ given that $$\displaystyle \int_{{ - 4}}^{0}{{f\left( x \right)\,dx}} = - 2$$, $$\displaystyle \int_{{31}}^{0}{{f\left( x \right)\,dx}} = 19$$ and $$\displaystyle \int_{{20}}^{{31}}{{f\left( x \right)\,dx}} = - 21$$. Solution

For problems 8 & 9 sketch the graph of the integrand and use the area interpretation of the definite integral to determine the value of the integral.

1. $$\displaystyle \int_{1}^{4}{{3x - 2\,dx}}$$ Solution
2. $$\displaystyle \int_{0}^{5}{{ - 4x\,dx}}$$ Solution

For problems 10 – 12 differentiate each of the following integrals with respect to x.

1. $$\displaystyle \int_{4}^{x}{{9{{\cos }^2}\left( {{t^2} - 6t + 1} \right)\,dt}}$$ Solution
2. $$\displaystyle \int_{7}^{{\sin \left( {6x} \right)}}{{\sqrt {{t^2} + 4} dt}}$$ Solution
3. $$\displaystyle \int_{{3{x^{\,2}}}}^{{ - 1}}{{\frac{{{{\bf{e}}^t} - 1}}{t}dt}}$$ Solution