5.4 Fundamental Theorem of Calculus

5.4 Fundamental Theorem of Calculus
• It is difficult to overestimate the power of the
• It says that every continuous function f is the
derivative of some other function, namely
It says that every continuous function has an
• It says that the processes of integration and
differentiation are inverses of one another.
Applying the Fundamental Theorem
• Find
Fundamental Theorem.
by using the
The Fundamental Theorem with the Chain Rule
• Find dy/dx if
Variable Lower Limits of Integration
• Find dy/dx.
Constructing a Function with a Given Derivative and
• Find a function y = f(x) with derivative
that satisfies the condition f(3) = 5.
• Since y(3) = 0, we have only to add 5 to this
function to construct one with derivative tan x
whose value at x = 3 is 5:
• The second part of the Fundamental Theorem of
Calculus shows how to evaluate definite integrals
directly from antiderivatives.
Evaluating an Integral
• Evaluate
using an
Finding Area Using Antiderivatives
• Find the area of the region between the curve
y = 4 – x², 0≤ x ≤ 3, and the x-axis.
• Textbook – p. 302 – 303 # 1 – 26, 41 – 44.