Calculus Examples

Find the Area Between the Curves y=2x , y=0 , x=3
, ,
Step 1
Solve by substitution to find the intersection between the curves.
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Step 1.1
Eliminate the equal sides of each equation and combine.
Step 1.2
Divide each term in by and simplify.
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Step 1.2.1
Divide each term in by .
Step 1.2.2
Simplify the left side.
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Step 1.2.2.1
Cancel the common factor of .
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Step 1.2.2.1.1
Cancel the common factor.
Step 1.2.2.1.2
Divide by .
Step 1.2.3
Simplify the right side.
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Step 1.2.3.1
Divide by .
Step 1.3
Substitute for .
Step 1.4
The solution to the system is the complete set of ordered pairs that are valid solutions.
Step 2
The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.
Step 3
Integrate to find the area between and .
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Step 3.1
Combine the integrals into a single integral.
Step 3.2
Subtract from .
Step 3.3
Since is constant with respect to , move out of the integral.
Step 3.4
By the Power Rule, the integral of with respect to is .
Step 3.5
Simplify the answer.
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Step 3.5.1
Combine and .
Step 3.5.2
Substitute and simplify.
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Step 3.5.2.1
Evaluate at and at .
Step 3.5.2.2
Simplify.
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Step 3.5.2.2.1
Raise to the power of .
Step 3.5.2.2.2
Raising to any positive power yields .
Step 3.5.2.2.3
Cancel the common factor of and .
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Step 3.5.2.2.3.1
Factor out of .
Step 3.5.2.2.3.2
Cancel the common factors.
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Step 3.5.2.2.3.2.1
Factor out of .
Step 3.5.2.2.3.2.2
Cancel the common factor.
Step 3.5.2.2.3.2.3
Rewrite the expression.
Step 3.5.2.2.3.2.4
Divide by .
Step 3.5.2.2.4
Multiply by .
Step 3.5.2.2.5
Add and .
Step 3.5.2.2.6
Combine and .
Step 3.5.2.2.7
Multiply by .
Step 3.5.2.2.8
Cancel the common factor of and .
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Step 3.5.2.2.8.1
Factor out of .
Step 3.5.2.2.8.2
Cancel the common factors.
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Step 3.5.2.2.8.2.1
Factor out of .
Step 3.5.2.2.8.2.2
Cancel the common factor.
Step 3.5.2.2.8.2.3
Rewrite the expression.
Step 3.5.2.2.8.2.4
Divide by .
Step 4