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 \(x_{\,i}^*\).
- \( \displaystyle \int_{1}^{4}{{2x + 3\,dx}}\) Solution
- \( \displaystyle \int_{0}^{1}{{6x\left( {x - 1} \right)\,dx}}\) Solution
- 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 \( \displaystyle \int_{6}^{{11}}{{g\left( x \right)\,dx}} = 24\).
- \( \displaystyle \int_{{11}}^{6}{{9f\left( x \right)\,dx}}\) Solution
- \( \displaystyle \int_{6}^{{11}}{{6g\left( x \right) - 10f\left( x \right)\,dx}}\) Solution
- 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
- 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.
- \( \displaystyle \int_{1}^{4}{{3x - 2\,dx}}\) Solution
- \( \displaystyle \int_{0}^{5}{{ - 4x\,dx}}\) Solution
For problems 10 – 12 differentiate each of the following integrals with respect to x.