QuestionFind the area of the region bound by the following equations:
Studdy Solution
STEP 1
What is this asking? We need to find the area under the line between and , and above the x-axis (). Watch out! Don't forget that we're looking for an *area*, so our answer should always be positive!
STEP 2
1. Define the integral
2. Calculate the integral
3. Evaluate the boundaries
STEP 3
Alright, let's **set up our integral**!
We're looking for the area under the curve , so we'll integrate this function.
Our **boundaries** are from to , so our integral looks like this:
This integral represents the **area** we're trying to find!
STEP 4
Time to **calculate the integral**!
Remember the power rule for integration: .
Don't forget the constant of integration, , even though it will disappear when we evaluate the definite integral.
STEP 5
Applying the power rule, the integral of is .
The integral of is .
So, our **integrated expression** is:
STEP 6
Now, let's **evaluate** our result at the **upper and lower limits** of integration.
We plug in (the **upper limit**) and subtract the result when we plug in (the **lower limit**).
STEP 7
Plugging in , we get: Plugging in , we get:
STEP 8
Finally, subtract the **lower limit result** from the **upper limit result**:
Notice how the constant of integration, , cancels out!
This always happens with definite integrals.
STEP 9
The area of the region is **144 square units**!
Was this helpful?
