Questions: Use the definite integral to find the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval. f(x)=75-3x^2; [0,8] Set up the integral (or integrals) needed to compute this area. Use the smallest possible number of integrals. Select the correct choice below and fill in the answer boxes to complete your choice. A. B. C. For the interval [0,8], the area between the x-axis and f(x) is . (Type an integer or a simplified fraction.)

Use the definite integral to find the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval. f(x)=75-3x^2; [0,8] Set up the integral (or integrals) needed to compute this area. Use the smallest possible number of integrals. Select the correct choice below and fill in the answer boxes to complete your choice. A. B. C. For the interval [0,8], the area between the x-axis and f(x) is . (Type an integer or a simplified fraction.)
Transcript text: Use the definite integral to find the area between the $x$-axis and $f(x)$ over the indicated interval. Check first to see if the graph crosses the $x$-axis in the given interval. \[ f(x)=75-3 x^{2} ;[0,8] \] Set up the integral (or integrals) needed to compute this area. Use the smallest possible number of integrals. Select the correct choice below and fill in the answer boxes to complete your choice. A. B. C. For the interval $[0,8]$, the area between the $x$-axis and $f(x)$ is $\square$ $\square$. (Type an integer or a simplified fraction.)
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Solution

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Solution Steps

Step 1: Find where f(x) crosses the x-axis.

Set f(x) = 0 and solve for x:

75 - 3x² = 0 3x² = 75 x² = 25 x = ±5

Since we are looking at the interval [0, 8], we only consider x = 5. The graph crosses the x-axis at x = 5, which is within the given interval.

Step 2: Set up the integrals.

Since the graph crosses the x-axis at x = 5, we need two integrals to represent the area: one from 0 to 5, and another from 5 to 8. The function is positive from 0 to 5, and negative from 5 to 8. To calculate the area, we must take the absolute value of the integral from 5 to 8.

∫₀⁵ (75 - 3x²) dx + ∫₅⁸ |75 - 3x²| dx

To avoid the absolute value, we can rewrite the second integral as:

∫₀⁵ (75 - 3x²) dx - ∫₅⁸ (75 - 3x²) dx

Step 3: Calculate the integrals

∫₀⁵ (75 - 3x²) dx = [75x - x³]₀⁵ = (75(5) - 5³) - (0) = 375 - 125 = 250

∫₅⁸ (75 - 3x²) dx = [75x - x³]₅⁸ = (75(8) - 8³) - (75(5) - 5³) = (600 - 512) - (375 - 125) = 88 - 250 = -162

Final Answer

250 - (-162) = 412

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