The Fundamental Theorem of Calculus is used to evaluate an [[integral]]. The Fundamental Theorem of Calculus states $\int_a^b f'(x)dx = f(b) - f(a)$ where $f'(x)$ is the derivative of $f(x)$ and $a$ and $b$ are the limits of integration. To solve the integral for any limits $a$ and $b$, find the [[antiderivative]], or the function that would give you the derivative in the integral, plug in $a$ and $b$ and subtract. This notation is often expressed as $\int_a^b f'(x)dx = f(x)\Big|_a^b = f(b) - f(a)$ In cases where the function is negative, and you want the area bounded by the function and the x-axis, you may need to take the absolute value across the range in which the function is negative. ## Example What is the integral from $0$ to $2$ of $x^2dx$? Writing this out we have $\int_0^2 x^2dx$ The anti-derivative of $x^2$ is $1/3x^3$. Plugging everything in we get $\int_0^2 x^2dx = 1/3x^3 \Big|_0^2 = 1/3(2)^3 - 1/3(0)^3 = 8/3$