Find The Area Between The Curves Y^2=4x , Y=2x-4 | Mathway

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Calculus Examples Popular Problems Calculus Find the Area Between the Curves y^2=4x , y=2x-4 , Step 1Solve by substitution to find the intersection between the curves.Tap for more steps...Step 1.1Replace all occurrences of with in each equation.Tap for more steps...Step 1.1.1Replace all occurrences of in with .Step 1.1.2Simplify the left side.Tap for more steps...Step 1.1.2.1Simplify .Tap for more steps...Step 1.1.2.1.1Rewrite as .Step 1.1.2.1.2Expand using the FOIL Method.Tap for more steps...Step 1.1.2.1.2.1Apply the distributive property.Step 1.1.2.1.2.2Apply the distributive property.Step 1.1.2.1.2.3Apply the distributive property.Step 1.1.2.1.3Simplify and combine like terms.Tap for more steps...Step 1.1.2.1.3.1Simplify each term.Tap for more steps...Step 1.1.2.1.3.1.1Rewrite using the commutative property of multiplication.Step 1.1.2.1.3.1.2Multiply by by adding the exponents.Tap for more steps...Step 1.1.2.1.3.1.2.1Move .Step 1.1.2.1.3.1.2.2Multiply by .Step 1.1.2.1.3.1.3Multiply by .Step 1.1.2.1.3.1.4Multiply by .Step 1.1.2.1.3.1.5Multiply by .Step 1.1.2.1.3.1.6Multiply by .Step 1.1.2.1.3.2Subtract from .Step 1.2Solve for in .Tap for more steps...Step 1.2.1Move all terms containing to the left side of the equation.Tap for more steps...Step 1.2.1.1Subtract from both sides of the equation.Step 1.2.1.2Subtract from .Step 1.2.2Factor the left side of the equation.Tap for more steps...Step 1.2.2.1Factor out of .Tap for more steps...Step 1.2.2.1.1Factor out of .Step 1.2.2.1.2Factor out of .Step 1.2.2.1.3Factor out of .Step 1.2.2.1.4Factor out of .Step 1.2.2.1.5Factor out of .Step 1.2.2.2Factor.Tap for more steps...Step 1.2.2.2.1Factor using the AC method.Tap for more steps...Step 1.2.2.2.1.1Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .Step 1.2.2.2.1.2Write the factored form using these integers.Step 1.2.2.2.2Remove unnecessary parentheses.Step 1.2.3If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .Step 1.2.4Set equal to and solve for .Tap for more steps...Step 1.2.4.1Set equal to .Step 1.2.4.2Add to both sides of the equation.Step 1.2.5Set equal to and solve for .Tap for more steps...Step 1.2.5.1Set equal to .Step 1.2.5.2Add to both sides of the equation.Step 1.2.6The final solution is all the values that make true.Step 1.3Replace all occurrences of with in each equation.Tap for more steps...Step 1.3.1Replace all occurrences of in with .Step 1.3.2Simplify the right side.Tap for more steps...Step 1.3.2.1Simplify .Tap for more steps...Step 1.3.2.1.1Multiply by .Step 1.3.2.1.2Subtract from .Step 1.4Replace all occurrences of with in each equation.Tap for more steps...Step 1.4.1Replace all occurrences of in with .Step 1.4.2Simplify the right side.Tap for more steps...Step 1.4.2.1Simplify .Tap for more steps...Step 1.4.2.1.1Multiply by .Step 1.4.2.1.2Subtract from .Step 1.5The solution to the system is the complete set of ordered pairs that are valid solutions.Step 2Solve in terms of .Tap for more steps...Step 2.1Rewrite the equation as .Step 2.2Divide each term in by and simplify.Tap for more steps...Step 2.2.1Divide each term in by .Step 2.2.2Simplify the left side.Tap for more steps...Step 2.2.2.1Cancel the common factor of .Tap for more steps...Step 2.2.2.1.1Cancel the common factor.Step 2.2.2.1.2Divide by .Step 3Solve in terms of .Tap for more steps...Step 3.1Rewrite the equation as .Step 3.2Add to both sides of the equation.Step 3.3Divide each term in by and simplify.Tap for more steps...Step 3.3.1Divide each term in by .Step 3.3.2Simplify the left side.Tap for more steps...Step 3.3.2.1Cancel the common factor of .Tap for more steps...Step 3.3.2.1.1Cancel the common factor.Step 3.3.2.1.2Divide by .Step 3.3.3Simplify the right side.Tap for more steps...Step 3.3.3.1Divide by .Step 4The 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 5Integrate to find the area between and .Tap for more steps...Step 5.1Combine the integrals into a single integral.Step 5.2Multiply by .Step 5.3Split the single integral into multiple integrals.Step 5.4Since is constant with respect to , move out of the integral.Step 5.5By the Power Rule, the integral of with respect to is .Step 5.6Apply the constant rule.Step 5.7Since is constant with respect to , move out of the integral.Step 5.8Since is constant with respect to , move out of the integral.Step 5.9By the Power Rule, the integral of with respect to is .Step 5.10Substitute and simplify.Tap for more steps...Step 5.10.1Evaluate at and at .Step 5.10.2Evaluate at and at .Step 5.10.3Evaluate at and at .Step 5.10.4Simplify.Tap for more steps...Step 5.10.4.1Raise to the power of .Step 5.10.4.2Combine and .Step 5.10.4.3Cancel the common factor of and .Tap for more steps...Step 5.10.4.3.1Factor out of .Step 5.10.4.3.2Cancel the common factors.Tap for more steps...Step 5.10.4.3.2.1Factor out of .Step 5.10.4.3.2.2Cancel the common factor.Step 5.10.4.3.2.3Rewrite the expression.Step 5.10.4.3.2.4Divide by .Step 5.10.4.4Raise to the power of .Step 5.10.4.5Multiply by .Step 5.10.4.6Combine and .Step 5.10.4.7Cancel the common factor of and .Tap for more steps...Step 5.10.4.7.1Factor out of .Step 5.10.4.7.2Cancel the common factors.Tap for more steps...Step 5.10.4.7.2.1Factor out of .Step 5.10.4.7.2.2Cancel the common factor.Step 5.10.4.7.2.3Rewrite the expression.Step 5.10.4.7.2.4Divide by .Step 5.10.4.8Subtract from .Step 5.10.4.9Combine and .Step 5.10.4.10Cancel the common factor of and .Tap for more steps...Step 5.10.4.10.1Factor out of .Step 5.10.4.10.2Cancel the common factors.Tap for more steps...Step 5.10.4.10.2.1Factor out of .Step 5.10.4.10.2.2Cancel the common factor.Step 5.10.4.10.2.3Rewrite the expression.Step 5.10.4.10.2.4Divide by .Step 5.10.4.11Multiply by .Step 5.10.4.12Multiply by .Step 5.10.4.13Add and .Step 5.10.4.14Add and .Step 5.10.4.15Raise to the power of .Step 5.10.4.16Combine and .Step 5.10.4.17Raise to the power of .Step 5.10.4.18Multiply by .Step 5.10.4.19Combine and .Step 5.10.4.20Combine the numerators over the common denominator.Step 5.10.4.21Add and .Step 5.10.4.22Cancel the common factor of and .Tap for more steps...Step 5.10.4.22.1Factor out of .Step 5.10.4.22.2Cancel the common factors.Tap for more steps...Step 5.10.4.22.2.1Factor out of .Step 5.10.4.22.2.2Cancel the common factor.Step 5.10.4.22.2.3Rewrite the expression.Step 5.10.4.22.2.4Divide by .Step 5.10.4.23Multiply by .Step 5.10.4.24Combine and .Step 5.10.4.25Cancel the common factor of and .Tap for more steps...Step 5.10.4.25.1Factor out of .Step 5.10.4.25.2Cancel the common factors.Tap for more steps...Step 5.10.4.25.2.1Factor out of .Step 5.10.4.25.2.2Cancel the common factor.Step 5.10.4.25.2.3Rewrite the expression.Step 5.10.4.25.2.4Divide by .Step 5.10.4.26Subtract from .Step 6

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