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8 problems match “definite integrals”. Clear

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1. Average value of an odd function ●○○○○ 10 ptsFind the average value of $R(z) = \sin(2z)\,e^{1-\cos(2z)}$ on the interval $[-\pi, \pi]$.
2. Average value of a cubic ●○○○○ 10 ptsWhat is the average value of $x^3-x^2$ on the interval $2\le x\le 5$? Give your answer as a decimal.
3. Area of one rose petal ●●○○○ 20 ptsFind the area enclosed by a single loop of the polar curve $r=4\cos(3\theta)$. Round to two decimal places.
4. Average value of sine ●●○○○ 20 ptsCalculate the average value of $f(x) = \sin(x)$ on the interval $[0, \pi]$, and separately on the interval $[0, 2\pi]$. Round each answer to two decimal places.
5. Points where f equals its average value ●●○○○ 20 ptsLet $f(x) = (x-2)^2$. Find all values of $c$ such that $f(c)$ equals the average value of $f$ over the closed interval $[-1, 5]$. Round each answer to two decimal places.
6. RMS voltage of household current ●●●○○ 50 ptsHousehold electricity has voltage $E(t)=155\sin(120\pi t)$ volts, where $t$ is time in seconds, and one cycle lasts $\frac{1}{60}$ second. The RMS voltage is the square root of the average value of $[E(t)]^2$ over one cycle. Find the RMS voltage of household current in volts. Round to one decimal place.
7. Average value of a trigonometric function ●●●○○ 50 ptsConsider the function $f(x) = 4\sin x - 2\sin 2x$ on the interval $[0, \pi]$. (a) Find the average value $f_{\text{ave}}$ of $f$ on this interval. (b) Find the two values $c_1 < c_2$ in $[0, \pi]$ such that $f(c_1) = f(c_2) = f_{\text{ave}}$. Round your answers to three decimal places.
8. Minimum ratio of derivative energy to squared mean ●●●●○ 100 ptsLet $f$ be a continuously differentiable real-valued function on $[0,1]$ with $\int_{1/3}^{2/3} f(x)\,dx = 0$. Find the minimum value of $\dfrac{\int_0^1 (f'(x))^2\,dx}{\left(\int_0^1 f(x)\,dx\right)^2}$ over all such $f$ for which the denominator is nonzero.