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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

At a Glance - How to Solve a Math Problem

There are three steps to solving a math problem.

1) Figure out what the problem is asking.

2) Solve the problem.

3) Check the answer.

Sample Problem

Lou paints at a rate of (140 – kt) square feet per hour, where t is the number of hours since he started painting and k is a positive constant accounting for the fact that Lou slows down as he gets tired. If Lou takes 6 hours to paint 500 square feet, how long does it take him to paint the last 100 square feet?


1) Figure out what the problem is asking.

The problem asks how long Lou takes to paint the last 100 square feet. We know he finishes painting when t = 6. We need to know when he started painting those last 100 square feet—that is, for what value of T is

This is an equation involving an integral. Unfortunately, it has letters both in the integrand and in the limits of integration:

If it weren't for that pesky "k" in the integrand we could work out the integral and solve the equation. This means we need to figure out what k is. Thankfully, the problem tells us enough to do that. We know Lou takes 6 hours to paint 500 square feet, so

We can solve this equation for k, then go back to the first equation and find T.

2) Solve the problem.

First we find k:

We need to use k for the next part, so we keep the exact answer

instead of rounding.

Now that we know k, we can solve the equation that will tell us the time at which Lou started painting the last 100 square feet:

Rearranging, we get a (horrible) quadratic equation:

Let's multiply through by 9 to make it slightly less horrible:

0 = 85T2 – 1260T + 3600.

Still horrible, but at least now we can apply the quadratic formula without worrying about having fractional coefficients.

The possible answers are


Since Lou finished his painting in 6 hours, he couldn't have started painting his last 100 square feet after 10.959 hours. This means the other number must be the one we want. It's reassuring that 3.865 is in between 0 and 6. Rounding to the nearest minute, we have

T ≈ 3.865(60) = 232 minutes,

or 3 hours and 52 minutes.

To finish, we need to answer the question that was actually asked, which was "how long" it takes Lou to paint those last 100 square feet. He started at 3 hours and 52 minutes, and finished at 6 hours, so it took him 2 hours and 8 minutes to paint the last 100 square feet.

3) Check the answer.

We've already done some checking along the lines of "does this answer make sense?" We know Lou painted for 6 hours and we found that he started painting the last 100 square feet after more than 0 but fewer than 6 hours, which is good.

Since this problem mostly consisted of finding values for k and T that made certain integrals come out to specific values, one way to check our answers is to take the values we found for k and T, stick the integrals into a calculator, and make sure they come out as they're supposed to.

We know the integral

is supposed to come out to 500, and we found . Stick

this value in for k, and put the resulting integral in your calculator:

You should get 500, which means we found the correct value of k.

Next, we need to see if we found the right value of T to make

come out to 100. We said Lou started painting his last 100 square feet at 232 minutes, or  hours. Since we rounded to the nearest minute this isn't an exact start time, which means when we plug in  for T we expect the integral to come out close to but not exactly equal to 100. Stick the following integral into your calculator:

We get about 99.87. You might get a slightly different answer, but it should still be close enough to 100 to convince you that we found the correct value of T.

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