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Part 2 (The Evaluation Theorem) Fundamental Theorem, Theorem of Calculus. This part describes the second part of the Fundamental Theorem of Calculus. We now introduce definite integrals without having to evaluate upper and lower limits of integration. We find and evaluate an antiderivative at the upper limit. The Fundamental Theorem of Calculus; Part 2 THEOREM 4 (Continued) – antiderivative of f on [a,b] then If f is continuous over and F is an antiderivative of f, then âˆ«f(x) dx = F(b) – F(a).

Part of the Fundamental Theorem tells us that an antiderivative of f exists.

Proof:

âˆ«f(t) dt.

Thus, if F is any antiderivative of f, then F(x) = G(x) + C for some constant C (by Corollary 2 of the Mean Value Theorem for Derivatives, Section). Since both F and G are continuous on [a,b], we see that the equality F(x) = G(x) also holds when x approaches a and b by taking one-sided limits (as x approaches a and b). Evaluating F(b) – F(a), we have

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02:46

The Fundamental Theorem of Calculus; Part 2If f is continuous over the interval [a, b] and F is any antiderivative of f(x), thenâˆ«(a to b) f(x) dx = F(b) – F(a).The value of âˆ«(a to b) f(x) dx is

01:19

‘The Fundamental Theorem of Calculus; Part 1If f is continuous on [a, b], then the function g defined byI)6 f(t) dt @ < 1 < b is continuous on [a, b] and differentiable on (a, b), and g (w) = f(o):We abbreviate the name of this theorem as FTCl. In words, it says that the derivative of a definiteintegral with respect to its upper limit is the integrand evaluated at the upper limit:’

06:12

Recall the Fundamental Theorem of Calculus _ which states that if f is continuous on [2_ b], then the following holds_ where is any antiderivative of That is_ F”x) f(x)f(x) dx = F(b) F(a)The Fundamental Theorem of Calculus can be used to find the area of the region under the graph of the function f(x) 2x on the interval [2, b] [-1, 2] The area of the region is given by the following.A =fI( 2x + 92r +Step 2To evaluate the definite integral, we first find an antiderivative F of f(x) 2X 9. Doing so gives the following result; where C is an arbitrary constant:F(x)In the next step we will evaluate the definite integral: Recall that the constant of integration will “drop out” when this computation is performed_ SO we may drop the C at this point: Doing So gives the following result:F(x) =Step

01:56

The Fundamental Theorem of Calculus (Part 2) states that if f(x) is a continuous function on an interval [a, b], then âˆ« f(x) dx = F(b) – F(a), where F(x) is an antiderivative of f(x).

The given function, f(x) = 50x^(-3), is defined for all x > 0 on the interval [3, 4] and is therefore continuous. Find an antiderivative F(x) of f(x).

F(x) = âˆ« 50x^(-3) dx

600

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