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### Course: AP®︎/College Calculus AB>Unit 6

Lesson 4: The fundamental theorem of calculus and accumulation functions

# Functions defined by definite integrals (accumulation functions)

Understanding that a function can be defined using a definite integral. Thinking about how to evaluate functions defined this way.

## Want to join the conversation?

• why do we end with, let's say, dx?
• Imagine a curve f(x) with a series of rectangles underneath it. The height of the rectangle would be f(x) and let us call the width of the rectangle delta x. The thinner you make your rectangles, the more rectangles you can fit within the area under the curve. Now, lets assume that each rectangle had an infinitesimally thin width, which we will call dx. Now, you get a perfect answer to the area under the curve. So the area of each rectangle is equal to f(x) dx (where dx is the width of the super thin rectangles). Now to find the total area under the curve, you have to add up all the areas of the rectangles which is denoted by ∫f(x) dx. Hope this helps! (Note: that ∫ is essentially an elongated "S" which stands for sum).
• Wouldn't the areas cancel out since one side is negative and the other side is positive?
• Nope, because even when t is -ve the area under the graph is in +ve y-axis so th areas would simply add up.
• What is g(-3). Is it a positive or a negative number?
• If the x-axis and the t-axis is the same axis, aren't "t" and "x" the same thing? Why don't we just use one or the other in the integral expression?
• why 21? the height is 4 and the width is 5, so I believe the integral should be 20.
• You're forgetting about the little triangle-shaped section on top of the rectangle.
• When the graph is under x axis will that value be negative?
• Yes. In the instances you want it to be positive you have to find each spot it crosses the x axis as solve each section seperately
• Why was -2, at , used as the lower boundary in the definite integral to define g(1)? It seems quite arbitrary.
• Yes, because it is just an example of how you can do it. It could also be 0 or 1 or even a function of `x`.
• Why is it that the definite integral from a to b of f(t) with respect to t is always the area underneath the curve from t=a to t=b?
• If you haven't watched the parts on riemann sums thse explain it. But basically the riemann sums split the area underneathe a graph into smaller and smaller rectangles, until they fit the graph exactly and thus give the exact area. the summation formula that is used is then used for integrals.