Example 2: Area Between Two Curves (Calculator Allowed)

What is the area enclosed by the curves $y=x^3–8x^2+18x–5$ and $y=x+5$.

Step 1: Determine whether your equation is in terms of x (standard: $y=x^2+3x+\sqrt{x}$) or in terms of y (non-standard: $x=y^2+3y+\sqrt{y}$).

In this example both of the equations are in terms of x .

$y=x^3–8x^2+18x–5$

$y=x+5$

Step 2: Determine the bounds of the integral, $\underset{a}{\overset{b}{\int }}$, for each of your enclosed regions.

In this example you are allowed to use a calculator. The best way to determine the bounds of your integral will be to graph the equations and calculate the intersections of the graphs.

– To do this you will put one of the equations into Y1 on our calculator and the other equation into Y2 .

– Then hit the Graph button.

You may need to adjust the Window of your graph to make sure you can see all the possible regions.

In this situation I would want to move the Y-Max up a little bit since both my graphs are leaving the screen on the top of the Window. In this example you just want to move the Y-Max up a little since we can almost see everything. I would adjust the Y-Max to something around y = 30 .

In this example you have two regions.

This means you will need to setup two different Definite Integrals, and then add their results to get the final answer.

You will need to identify the bounds for each different region. The most straight forward way to find the bounds will be to use the Calculate Intersect function on our calculator.

You will need to apply this process three different times to find your three different intersections , the three different bounds you need to setup your Definite Integral.

Calculate Intersect Directions:

– Hit

– Then

– Then choose the #5 Option: Intersect.

From there you will always be asked a series of three questions.

– Question 1: First Curve?

It will ask you to identify the First Curve you want to use in the intersect process.

You just want to hit .

– Question 2: Second Curve?

It will then ask you to identify the Second Curve you want to use in the intersect process.

You just want to hit .

Note: Most of the time you will only have two curves so you will always hit Enter, Enter, through the first two questions.

– Question 3: It will ask you to Guess?

Which means it wants you to move the cursor close to the intersection you are trying to calculate using the left and right arrows. The calculator will always find the intersection it is closest to , so this is you telling the calculator the intersection you are specifically trying to calculate.

How you answer this third question will be the only part of the process that changes from one intersection to the next. Remember you will have to run these steps multiple times to find all your intersections , all of your bounds .

Once you are close to the intersection you want to find hit a third time.

Region 1:

Lower bound, a = 1

Upper bound, b = 2

Region 2:

Lower bound, a = 2

Upper bound, b = 5

2 ^nd :

Calc:

#5 Option: Intersection:

First Curve:

Second Curve:

Guess Lower Bound Region 1:

Guess Upper Bound Region 1 = Lower Bound Region 2

Guess Upper Bound Region 2:

Intersect Lower Bound Region 1:

Intersect Upper Bound Region 1 = Lower Bound Region 2

Intersect Upper Bound Region 2:

Bounds for Region 1: $\underset{a}{\overset{b}{\int }}=\underset{1}{\overset{2}{\int }}$

Bounds for Region 2: $\underset{a}{\overset{b}{\int }}=\underset{2}{\overset{5}{\int }}$

Step 3: Determine the Top and Bottom equations for all of your enclosed regions.

In this example you are allowed to use a calculator. The best way to determine your Top and Bottom will be to graph the equations.

You already have the graph in your calculator at this point, and really this is just a matter of reading the information off of that graph.

In your example you have two regions.

This means we will need to identify the Top and Bottomfor each different region. You will need to setup a Definite Integrals for each reason and then add their results to get the final answer.

In Region 1: the Blue graph is on Top and the Red graph is on Bottom.

Top :  $y=x^3–8x^2+18x–5$

Bottom :   $y=x+5$

In Region 2: the graphs have flipped position so Red graph is on Top and the Blue graph is on Bottom.

Top :  $y=x+5$

Bottom:   $y=x^3–8x^2+18x–5$

Step 4: Setup a Definite Integral problem for each of your enclosed regions using your Area Between Two Curves formula :

$\underset{a}{\overset{b}{\int }}\mathit{Top}–\left(\mathit{Bottom}\right)\mathit{dx}=\mathit{Area\; Betwen }2\mathit{ Curves}$

Evaluating the Definite Integrals:

The great thing about being able to use your calculator is that you can use the calculator to find the answer to these Definite Integral problems very easily.

Region 1 Calculator Steps

From Step 2: $\underset{1}{\overset{2}{\int }}$

From Step 3:

Top :  $y=x^3–8x^2+18x–5$

Bottom :   $y=x+5$

$\underset{1}{\overset{2}{\int }}{x}^3–8x^2+18x–5–\left(x+5\right)\mathit{dx}=\mathit{Area\; Betwen }2\mathit{ Curves}$

Enter your equation into Y1:

Region 2 Calculator Steps

From Step 2: $\underset{2}{\overset{5}{\int }}$

From Step 3:

Top :  $y=x+5$

Bottom:   $y=x^3–8x^2+18x–5$

$\underset{2}{\overset{5}{\int }}x+5–\left(x^3–8x^2+18x–5\right)\mathit{dx}=\mathit{Area\; Betwen }2\mathit{ Curves}$

Enter your equation into Y1:

Hit , then  — , and choose the #7 Option .

You are then asked to input the “ Lower Limit ”.

You will want to type in your Lower Bound for Region 1, which would be a = 1 and then hit

.

You are then asked to input the “ Lower Limit ”.

You will want to type in your Lower Bound for Region 2, which would be a = 2 and then hit

.

You are then asked to input the “ Upper Limit ”.

You will want to type in your Upper Bound for Region 1, which would be b = 2 and then hit

.

You are then asked to input the “ Upper Limit ”.

You will want to type in your Upper Bound for Region 2, which would be b = 5 and then hit

.

The calculator will then shade in Region 1’s area on the screen, and at the bottom of the graph it will output the answer.

The Final Result for Region 1 would be:

$\underset{1}{\overset{2}{\int }}{x}^3–8x^2+18x–5–\left(x+5\right)\mathit{dx}\approx .58333$

The calculator will then shade in Region 2’s area on the screen, and at the bottom of the graph it will output the answer.

The Final Result for Region 2 would be:

$\underset{2}{\overset{5}{\int }}x+5–\left(x^3–8x^2+18x–5\right)\mathit{dx}=11.25$

Final Result:

The area enclosed by the curves $y=x^3–8x^2+18x–5$ and $y=x+5$is found by adding the area enclosed by the two regions the curves combine to create.

$\underset{1}{\overset{2}{\int }}{x}^3–8x^2+18x–5–\left(x+5\right)\mathit{dx}+\underset{2}{\overset{5}{\int }}x+5–\left(x^3–8x^2+18x–5\right)\mathit{dx}=\mathit{Total\; Area\; Betwen }2\mathit{ Curves}$

$58333+11.25\approx 11.83333\approx \mathit{Total\; Area\; Betwen }2\mathit{ Curves}$