V=piint_(-1)^(2)[x+2]^(2)-[x^(2)]^(2)dx

Expand functions: We are asked to evaluate the definite integral of the function (x+2)^2 - (x^2)^2 from x = -1 to x = 2. The first step is to expand the functions inside the integral before integrating.Substitute expanded forms: Expand (x+2)^2: (x+2)^2 = x^2 + 4x + 4Separate into individual terms: Expand (x^2)^2: (x^2)^2 = x^4Integrate each term: Substitute the expanded forms into the integral: V = pi * ∫ from -1 to 2 of (x^2 + 4x + 4 - x^4) dxApply limits of integration: Separate the integral into individual terms: V = pi * (∫ from -1 to 2 of x^2 dx + ∫ from -1 to 2 of 4x dx + ∫ from -1 to 2 of 4 dx - ∫ from -1 to 2 of x^4 dx)Evaluate at upper and lower limits: Integrate each term separately: ∫ of x^2 dx = (1/3)x^3 ∫ of 4x dx = 2x^2 ∫ of 4 dx = 4x ∫ of x^4 dx = (1/5)x^5Perform calculations: Apply the limits of integration to each term: V = pi * ([(1/3)x^3] from -1 to 2 + [2x^2] from -1 to 2 + [4x] from -1 to 2 - [(1/5)x^5] from -1 to 2)Simplify the expression: Evaluate each term at the upper and lower limits and subtract: V = pi * ((1/3)(2)^3 - (1/3)(-1)^3 + 2(2)^2 - 2(-1)^2 + 4(2) - 4(-1) - (1/5)(2)^5 + (1/5)(-1)^5)Multiply by pi: Perform the calculations: V = pi * ((1/3)(8) - (1/3)(-1) + 2(4) - 2(1) + 8 - (-4) - (1/5)(32) + (1/5)(-1))Simplify the expression: V = pi * ((8/3) + (1/3) + 8 - 2 + 8 + 4 - (32/5) - (1/5)) V = pi * ((9/3) + 6 + 12 - (33/5)) V = pi * (3 + 6 + 12 - (33/5)) V = pi * (21 - (33/5)) V = pi * (105/5 - 33/5) V = pi * (72/5)Multiply by pi to get the final answer: V = pi * (72/5) V = (72/5)pi

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$V=\pi \int_{-1}^{2}[x+2]^{2}-\left[x^{2}\right]^{2} d x$

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Find derivatives of using multiple formulae

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V=\pi \int_{-1}^{2}[x+2]^{2}-\left[x^{2}\right]^{2} d x

Expand functions: We are asked to evaluate the definite integral of the function $(x+2)^2 - (x^2)^2$ from $x = -1$ to $x = 2$ . The first step is to expand the functions inside the integral before integrating.
Substitute expanded forms: Expand $(x+2)^2$ : $(x+2)^2 = x^2 + 4x + 4$
Separate into individual terms: Expand $(x^2)^2$ : $(x^2)^2 = x^4$
Integrate each term: Substitute the expanded forms into the integral: $V = \pi \cdot \int_{-1}^{2} (x^2 + 4x + 4 - x^4) \, dx$
Apply limits of integration: Separate the integral into individual terms: $\newline$ $V = \pi \cdot \left(\int_{-1}^{2} x^2 \, dx + \int_{-1}^{2} 4x \, dx + \int_{-1}^{2} 4 \, dx - \int_{-1}^{2} x^4 \, dx\right)$
Evaluate at upper and lower limits: Integrate each term separately: $\newline$ $\int x^2 \, dx = \frac{1}{3}x^3$ $\newline$ $\int 4x \, dx = 2x^2$ $\newline$ $\int 4 \, dx = 4x$ $\newline$ $\int x^4 \, dx = \frac{1}{5}x^5$
Perform calculations: Apply the limits of integration to each term: $V = \pi \cdot (\left[\frac{1}{3}x^3\right]_{-1}^{2} + \left[2x^2\right]_{-1}^{2} + \left[4x\right]_{-1}^{2} - \left[\frac{1}{5}x^5\right]_{-1}^{2})$
Simplify the expression: Evaluate each term at the upper and lower limits and subtract: $V = \pi \cdot \left(\frac{1}{3}(2)^3 - \frac{1}{3}(-1)^3 + 2(2)^2 - 2(-1)^2 + 4(2) - 4(-1) - \frac{1}{5}(2)^5 + \frac{1}{5}(-1)^5\right)$
Multiply by pi: Perform the calculations: $\newline$ $V = \pi \times \left(\left(\frac{1}{3}\right)(8) - \left(\frac{1}{3}\right)(-1) + 2(4) - 2(1) + 8 - (-4) - \left(\frac{1}{5}\right)(32) + \left(\frac{1}{5}\right)(-1)\right)$
Multiply by pi: Perform the calculations: $\newline$ $V = \pi \times \left(\left(\frac{1}{3}\right)(8) - \left(\frac{1}{3}\right)(-1) + 2(4) - 2(1) + 8 - (-4) - \left(\frac{1}{5}\right)(32) + \left(\frac{1}{5}\right)(-1)\right)$ Simplify the expression: $\newline$ $V = \pi \times \left(\left(\frac{8}{3}\right) + \left(\frac{1}{3}\right) + 8 - 2 + 8 + 4 - \left(\frac{32}{5}\right) - \left(\frac{1}{5}\right)\right)$ $\newline$ $V = \pi \times \left(\left(\frac{9}{3}\right) + 6 + 12 - \left(\frac{33}{5}\right)\right)$ $\newline$ $V = \pi \times \left(3 + 6 + 12 - \left(\frac{33}{5}\right)\right)$ $\newline$ $V = \pi \times \left(21 - \left(\frac{33}{5}\right)\right)$ $\newline$ $V = \pi \times \left(\frac{105}{5} - \frac{33}{5}\right)$ $\newline$ $V = \pi \times \left(\frac{72}{5}\right)$
Multiply by pi: Perform the calculations: $\newline$ $V = \pi \times \left(\left(\frac{1}{3}\right)(8) - \left(\frac{1}{3}\right)(-1) + 2(4) - 2(1) + 8 - (-4) - \left(\frac{1}{5}\right)(32) + \left(\frac{1}{5}\right)(-1)\right)$ Simplify the expression: $\newline$ $V = \pi \times \left(\left(\frac{8}{3}\right) + \left(\frac{1}{3}\right) + 8 - 2 + 8 + 4 - \left(\frac{32}{5}\right) - \left(\frac{1}{5}\right)\right)$ $\newline$ $V = \pi \times \left(\left(\frac{9}{3}\right) + 6 + 12 - \left(\frac{33}{5}\right)\right)$ $\newline$ $V = \pi \times \left(3 + 6 + 12 - \left(\frac{33}{5}\right)\right)$ $\newline$ $V = \pi \times \left(21 - \left(\frac{33}{5}\right)\right)$ $\newline$ $V = \pi \times \left(\frac{105}{5} - \frac{33}{5}\right)$ $\newline$ $V = \pi \times \left(\frac{72}{5}\right)$ Multiply by pi to get the final answer: $\newline$ $V = \pi \times \left(\frac{72}{5}\right)$ $\newline$ $V = \left(\frac{72}{5}\right)\pi$

$V=\pi \int_{-1}^{2}[x+2]^{2}-\left[x^{2}\right]^{2} d x$

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V=π∫−12[x+2]2−[x2]2dx V=\pi \int_{-1}^{2}[x+2]^{2}-\left[x^{2}\right]^{2} d x V=π∫−12​[x+2]2−[x2]2dx

$V=\pi \int_{-1}^{2}[x+2]^{2}-\left[x^{2}\right]^{2} d x$