Find the area under the curve, of the function $f\left(x\right)=x^2$ from $x=0$ to $x=8$ using limits.

• A. 256
• B. $512$
• C. $\frac{1}{3}$
• D. $\frac{512}{3}$

Answer: (D)

Verified step by step guidance

1. To find the area under the curve of the function \( f(x) = x^2 \) from \( x = 0 \) to \( x = 8 \), we need to calculate the definite integral of \( f(x) \) over the interval \([0, 8]\).
2. The definite integral of a function \( f(x) \) from \( a \) to \( b \) is given by \( \int_{a}^{b} f(x) \, dx \). In this case, it is \( \int_{0}^{8} x^2 \, dx \).
3. To evaluate this integral, we first find the antiderivative of \( x^2 \). The antiderivative of \( x^2 \) is \( \frac{x^3}{3} \).
4. Next, we apply the Fundamental Theorem of Calculus, which states that \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \), where \( F(x) \) is the antiderivative of \( f(x) \).
5. Substitute \( x = 8 \) and \( x = 0 \) into the antiderivative \( \frac{x^3}{3} \) and compute \( \frac{8^3}{3} - \frac{0^3}{3} \) to find the area under the curve.