Math Snap
PROBLEM
oblem
oblem
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oblem 4
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lem 9
STEP 1
What is this asking?
We need to find the area under the line from to .
Watch out!
Don't forget that area below the x-axis counts as negative!
STEP 2
1. Find the antiderivative.
2. Evaluate the antiderivative at the boundaries.
3. Calculate the definite integral.
STEP 3
Let's find the antiderivative of our function .
Remember, the power rule for integration says that the antiderivative of is .
STEP 4
The antiderivative of (which is the same as ) is .
We multiplied by raised to the old power plus one, and then divided by the new power!
STEP 5
The antiderivative of (which is the same as ) is .
STEP 6
So, the complete antiderivative of is .
Don't forget the constant of integration, which we can write as .
Since we're doing a definite integral, the will disappear later, but it's good practice to include it!
STEP 7
Now, we evaluate our antiderivative at the upper limit of integration, which is .
Substituting into , we get .
STEP 8
Next, we evaluate at the lower limit of integration, .
Substituting into , we get .
STEP 9
To find the definite integral, we subtract the value of the antiderivative at the lower limit from the value at the upper limit.
STEP 10
So, we have .
See how the and add to zero?
This always happens with definite integrals!
SOLUTION
The final answer, the definite integral of from to , is , or !