Area Between Two Curves (Calc 1 Problem Solution Breakdown)

Finding the area under a curve is a basic Calc 1 concept, but finding the area between two curves is a bit more difficult.

In today’s video, I go through a problem breakdown from a Calc 1 exam where we’re asked to find the area between two curves.

Here’s how to do it, step by step.

Area Between Two Curves Problem Statement

Let f(x) = x^2 - 4*x + 3. The graph of y = f(x) is shown below.

(a) Compute  \int_0^5f(x) dx .

(b) Find the total area of the shaded region.



area between two curves image 1


area between two curves image 2


0:36 – Breaking down the problem statement: In most cases where you have a function that is above the x axis the entire time, it’s pretty straightforward to find the area under the curve. To do that you just set up a definite integral and solve. Here what we’re asked to do is evaluate the area under the curve, but our function here crosses the x axis, so it’s a little more tricky.

1:05 – Dissecting how we got the final answer for the total area: The final answer is this 28 over 3. By looking at step above, we see that we have three absolute values. Why are we taking the absolute value and why is it broken up into three different values that were added together?

1:36 – Dissecting why 3 separate integrals and absolute values were used to find the total area: The only area we’re seeing that is negative is this second area (-4/3). That’s the only time that the absolute value makes a difference. Just by looking at this, we can associate that we’re getting a positive area when we take the integral when the function is above the x axis, but then we’re getting a negative area when we take the integral when the function is below the x axis.

3:08 – How to find where to split up your integrals: If we have a function that crosses the x axis, we need to find out what those x values are (the zeros). Then we can separate our integrals into multiple integrals when the function reaches that x value.

4:34 – Finding the net integral of the bounded area using a definite integral: It’s going to be the same as just adding these three values together without taking the absolute value. We did that by taking the antiderivative of f(x) and then plugged in the two bounds, 5 and 0.

6:30 – Analyzing the difference between net area and total area: I thought integrals find the area under the curve, so why doesn’t this represent the “total area” in this case? When your area that you’re finding is above the x axis, it ends up being positive. When it’s below, it ends up being negative. When you’re taking the integral from 0 to 5 here, what you’re actually finding is the net area. That’s just summing up the positives and the negatives to get the net, vs. the total area you’re finding the overall absolute value of the area trapped between these two curves.

Problem taken from:

Area Between Two Curves Concept Explanation (Paul’s Online Math Notes):

That’s a Reverse Learning breakdown of finding the area between two curves. Hopefully this helps your understanding of how to set up and complete these types of problems.

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