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