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How do you evaluate $\int_{a}^{b} f(x) dx$ using FTOC Part 2?

Find the antiderivative F(x) of f(x). 2. Evaluate F(b) and F(a). 3. Subtract F(a) from F(b): F(b) - F(a).

How do you find $g'(x)$ if $g(x) = \int_{a}^{x} f(t) dt$ ?

Apply FTOC Part 1: $g'(x) = f(x)$ .

How do you handle a constant of integration when using FTOC Part 2?

The constant of integration cancels out when evaluating F(b) - F(a), so it's not necessary to include it.

How to find $g'(x)$ if $g(x) = \int_{a}^{h(x)} f(t) dt$ ?

Apply FTOC Part 1 and the chain rule: $g'(x) = f(h(x)) * h'(x)$

Define definite integral.

An integral with upper and lower limits, resulting in a numerical value.

What is an antiderivative?

A function whose derivative is the original function.

Define the Fundamental Theorem of Calculus.

A theorem that connects the derivative and the integral, stating that differentiation and integration are inverse processes.

What does the Fundamental Theorem of Calculus, Part 1 state?

If $g(x) = \int_{a}^{x} f(t) dt$ , then $g'(x) = f(x)$ .

What does the Fundamental Theorem of Calculus, Part 2 state?

If $F(x)$ is an antiderivative of $f(x)$ , then $\int_{a}^{b} f(x) dx = F(b) - F(a)$ .