Calculus Examples

Find the Area Between the Curves y=x^2 , y=4
,
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
Solve for .
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Step 1.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.2.2
Simplify .
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Step 1.2.2.1
Rewrite as .
Step 1.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 1.2.3
The complete solution is the result of both the positive and negative portions of the solution.
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Step 1.2.3.1
First, use the positive value of the to find the first solution.
Step 1.2.3.2
Next, use the negative value of the to find the second solution.
Step 1.2.3.3
The complete solution is the result of both the positive and negative portions of the solution.
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
Multiply by .
Step 3.3
Split the single integral into multiple integrals.
Step 3.4
Apply the constant rule.
Step 3.5
Since is constant with respect to , move out of the integral.
Step 3.6
By the Power Rule, the integral of with respect to is .
Step 3.7
Simplify the answer.
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Step 3.7.1
Combine and .
Step 3.7.2
Substitute and simplify.
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Step 3.7.2.1
Evaluate at and at .
Step 3.7.2.2
Evaluate at and at .
Step 3.7.2.3
Simplify.
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Step 3.7.2.3.1
Multiply by .
Step 3.7.2.3.2
Multiply by .
Step 3.7.2.3.3
Add and .
Step 3.7.2.3.4
Raise to the power of .
Step 3.7.2.3.5
Raise to the power of .
Step 3.7.2.3.6
Move the negative in front of the fraction.
Step 3.7.2.3.7
Multiply by .
Step 3.7.2.3.8
Multiply by .
Step 3.7.2.3.9
Combine the numerators over the common denominator.
Step 3.7.2.3.10
Add and .
Step 3.7.2.3.11
To write as a fraction with a common denominator, multiply by .
Step 3.7.2.3.12
Combine and .
Step 3.7.2.3.13
Combine the numerators over the common denominator.
Step 3.7.2.3.14
Simplify the numerator.
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Step 3.7.2.3.14.1
Multiply by .
Step 3.7.2.3.14.2
Subtract from .
Step 4