Calculus I (AB): Accumulation Functions

If , calculate f(2) and . Do not use a calculator.

Difficulty:

Solution:

The first order of business is to find f(2). Good news: That's easy. Simply plug in 2 for x (not for t) in the accumulation function:

On the other hand, is a bit more difficult to find. Since both of the boundaries of integration contain variables, you cannot use the cool shortcut offered by the Fundamental Theorem of Calculus (which basically says that derivatives and integrals cancel each other out). Instead, you actually have to integrate with respect to t, plug in the upper and lower bounds (even though they have x's), and subtract. In other words, the long way. In summary, before we can calculate the derivative of f, we need to come up with a polynomial representation of f:

How did I know when I got it right? This function should be equivalent to , so f(2) = 60.75 in both of them.

Finally, it's time to figure out . All you have to is differentiate the function we just worked so hard to create and plug in 1 for x:

Note: The above solution represents a very conceptual approach to the problem. Jeffrey Lockshin, a Calculus-Help.com reader, offers the following shortcut solution:

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Determine the value of the definite integral .

Difficulty:

Solution:

The infinite upper bound causes this integral to be improper, so we should replace it with a constant and take the limit as that constant approaches infinity, like so:

Remember that will approach 0 as b approaches infinity (since the denominator will get infinitely large).