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Math  /  Calculus

QuestionFind the total area bounded by the xx-axis and the curve y=f(x)y=f(x) on the indicated interval. Enter your answer in exact form or as a decimal number rounded to the nearest thousandth. f(x)=27x−8;[3,5]f(x)=2 \sqrt{7 x-8} ;[3,5]
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Studdy Solution

STEP 1

1. We are given the function f(x)=27x−8 f(x) = 2 \sqrt{7x - 8} .
2. We need to find the total area bounded by the x x -axis and the curve on the interval [3,5][3, 5].

STEP 2

1. Set up the definite integral to find the area under the curve.
2. Evaluate the integral.
3. Calculate the exact area or round to the nearest thousandth.

STEP 3

Set up the definite integral for the area under the curve from x=3 x = 3 to x=5 x = 5 :
A=∫3527x−8 dx A = \int_{3}^{5} 2 \sqrt{7x - 8} \, dx

STEP 4

To evaluate the integral, use substitution. Let u=7x−8 u = 7x - 8 .
Then, dudx=7 \frac{du}{dx} = 7 or dx=du7 dx = \frac{du}{7} .
Change the limits of integration: - When x=3 x = 3 , u=7(3)−8=13 u = 7(3) - 8 = 13 . - When x=5 x = 5 , u=7(5)−8=27 u = 7(5) - 8 = 27 .
The integral becomes:
A=∫13272u⋅17 du A = \int_{13}^{27} 2 \sqrt{u} \cdot \frac{1}{7} \, du
A=27∫1327u du A = \frac{2}{7} \int_{13}^{27} \sqrt{u} \, du

STEP 5

Evaluate the integral:
∫u du=∫u1/2 du=u3/23/2=23u3/2 \int \sqrt{u} \, du = \int u^{1/2} \, du = \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2}
Substitute back into the integral:
A=27[23u3/2]1327 A = \frac{2}{7} \left[ \frac{2}{3} u^{3/2} \right]_{13}^{27}
A=421[u3/2]1327 A = \frac{4}{21} \left[ u^{3/2} \right]_{13}^{27}

STEP 6

Calculate the definite integral:
A=421[(27)3/2−(13)3/2] A = \frac{4}{21} \left[ (27)^{3/2} - (13)^{3/2} \right]
Calculate each term:
- (27)3/2=(271/2)3=33=27 (27)^{3/2} = (27^{1/2})^3 = 3^3 = 27 - (13)3/2=(131/2)3=(13)3 (13)^{3/2} = (13^{1/2})^3 = (\sqrt{13})^3
A=421[27−(13)3] A = \frac{4}{21} \left[ 27 - (\sqrt{13})^3 \right]
Calculate (13)3 (\sqrt{13})^3 and then the area:
- 13≈3.606 \sqrt{13} \approx 3.606 - (13)3≈3.6063≈46.872 (\sqrt{13})^3 \approx 3.606^3 \approx 46.872
A=421[27−46.872] A = \frac{4}{21} \left[ 27 - 46.872 \right] A=421[−19.872] A = \frac{4}{21} \left[ -19.872 \right] A≈4×−19.87221 A \approx \frac{4 \times -19.872}{21} A≈−3.782 A \approx -3.782
Since area cannot be negative, take the absolute value:
A≈3.782 A \approx 3.782
Round to the nearest thousandth:
A≈3.782 A \approx 3.782
The total area bounded by the x x -axis and the curve is approximately:
3.782 \boxed{3.782}

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