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Hitunglah volume benda padat pada oktan $I$ yang dibatasi oleh silinder elip $y^2 +64z^2= 4$ dan bidang $y=z$

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Find derivatives using the quotient rule II

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Q. Hitunglah volume benda padat pada oktan

I

yang dibatasi oleh silinder elip

y^2 +64z^2= 4

dan bidang

y=z

Identify Bounds: Identify the bounds for integration. $\newline$ The elliptical cylinder $y^2 + 64z^2 = 4$ can be rewritten as $z^2 = \frac{4 - y^2}{64}$ . Since we're looking at the first octant, $y$ and $z$ are non-negative, and $y = z$ defines another boundary.
Set up Integral: Set up the integral for volume. $\newline$ The volume $V$ can be found by integrating the area of cross-sections. The cross-sections perpendicular to the x-axis are rectangles with one side along the y-axis from $0$ to $z$ and the other side along the z-axis from $0$ to $z$ . The length of each side is $z$ , so the area of each cross-section is $z^2$ .
Convert Equation: Convert the equation of the elliptical cylinder to express $z$ in terms of $y$ . $z^2 = \frac{4 - y^2}{64}$ leads to $z = \sqrt{\frac{4 - y^2}{64}}.$
Determine Limits: Determine the limits of integration for $y$ . Since $y = z$ and we are in the first octant, the limits for $y$ are from $0$ to the square root of $4$ , which is $2$ .
Write Integral: Write the integral to calculate the volume. $\newline$ $V = \int_{y=0}^{y=2} \left(\sqrt{\frac{4 - y^2}{64}}\right)^2 dy$ .
Simplify Integral: Simplify the integral. $V = \int_{y=0}^{y=2} \frac{4 - y^2}{64} dy$ .
Perform Integration: Perform the integration. $V = \left[\frac{y}{64} - \frac{y^3}{3\times64}\right]$ from $0$ to $2$ .
Evaluate Bounds: Evaluate the integral at the bounds. $\newline$ $V = \left[\left(\frac{2}{64}\right) - \left(\frac{2^3}{3\cdot64}\right)\right] - \left[\left(\frac{0}{64}\right) - \left(\frac{0^3}{3\cdot64}\right)\right]$ .
Simplify Result: Simplify the result. $\newline$ $V = \frac{2}{64} - \frac{8}{3 \times 64}$ .
Combine Terms: Combine the terms to find the volume. $V = \frac{1}{32} - \frac{1}{3 \times 32}$ .
Calculate Final Value: Calculate the final value. $V = \frac{3}{96} - \frac{1}{96} = \frac{2}{96} = \frac{1}{48}$ .

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