- Compute the following definite integrals.
- [latex]\int_{2}^3 x^3 + 2 \sqrt{x} dx[/latex]
19.04ans
- [latex]\int_{-2}^3 (x + 5)^2dx[/latex]
[latex]\approx 161.7[/latex]ans
- [latex]\int_{0}^{1} e^xdx[/latex]
[latex]e - 1 \approx 1.718[/latex]ans
- [latex]\int_{-1}^1 3 e^xdx[/latex]
[latex]7.05[/latex]ans
- [latex]\int_{1}^{e} \frac{3}{x} + \frac{x}{3}\ dx[/latex]
[latex]\approx 4.06[/latex]ans
- [latex]\int_{2}^3 x^3 + 2 \sqrt{x} dx[/latex]
- Approximate [latex]\int_0^1 x^2dx[/latex] using [latex]4[/latex] rectangles. Then find [latex]\int_0^1 x^2[/latex] exactly using an anti-derivative. How far off is the approximation?
Approximation [latex]\approx 0.22[/latex], the actual is [latex]\frac{1}{3} \approx .33[/latex], so the difference is about [latex]0.11[/latex] or [latex]50[/latex] error which isn't great. As we know, the rectangles don't always do such a good job.ans