Math / Calculus

QuestionCalculate $F(8)$ given that $F(5)=3$ and $F^{\prime}(x)=x^{2}$ . Hint: Express $F(8)-F(5)$ as a definite integral. (Use symbolic notation and fractions where needed.) $F(8)=$

Studdy Solution

STEP 1

1. $F(x)$ is a differentiable function.
2. $F'(x) = x^2$ is the derivative of $F(x)$ .
3. We are given $F(5) = 3$ .

STEP 2

1. Express $F(8) - F(5)$ as a definite integral.
2. Evaluate the definite integral.
3. Solve for $F(8)$ .

STEP 3

Express $F(8) - F(5)$ as a definite integral using the Fundamental Theorem of Calculus:
$F(8) - F(5) = \int_{5}^{8} F'(x) \, dx$
Since $F'(x) = x^2$ , substitute into the integral:
$F(8) - F(5) = \int_{5}^{8} x^2 \, dx$

STEP 4

Evaluate the definite integral:
$\int_{5}^{8} x^2 \, dx$
First, find the antiderivative of $x^2$ :
$\int x^2 \, dx = \frac{x^3}{3} + C$
Now, evaluate from 5 to 8:
$\left[ \frac{x^3}{3} \right]_{5}^{8} = \frac{8^3}{3} - \frac{5^3}{3}$
Calculate the values:
$\frac{8^3}{3} = \frac{512}{3}$ $\frac{5^3}{3} = \frac{125}{3}$
Subtract the results:
$\frac{512}{3} - \frac{125}{3} = \frac{387}{3} = 129$

STEP 5

Solve for $F(8)$ :
Given $F(5) = 3$ , we have:
$F(8) - 3 = 129$
Add 3 to both sides:
$F(8) = 129 + 3 = 132$
The value of $F(8)$ is:
$\boxed{132}$

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