Ly Ul Polht H, 380m Vertically Above A Lake. ULe O - Gauthmath

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Math Resources/Calculus/

Question

Find the area bounded by the curve y^2=4x and the line y+2x=12 A. 25.67 B. 37.71 C. 41.67 D. 56.17 10 Find the area bounded by the parabolas y^2=4x and x^2=4ySHOW LESS107

Solution

user avatar imageAnswer\(41.67\)ExplanationStep 1: Set up the system of equations to find the intersection points of the curve \(y^{2}=4x\) and the line \(y+2x=12\). Step 2: Solve the system of equations to find the intersection points. Substitute \(y^{2}=4x\) into the line equation to get \(y+2\left(\frac{y^{2}}{4}\right)=12\), which simplifies to \(y+y^{2}=12\). Step 3: Solve the quadratic equation \(y^{2}+y-12=0\) to find the values of \(y\). The solutions are \(y=4\) and \(y=-6\). Step 4: Substitute \(y=4\) into the equation \(y^{2}=4x\) to find the corresponding \(x\) value, which gives \(x=4\). Step 5: Substitute \(y=-6\) into the equation \(y^{2}=4x\) to find the corresponding \(x\) value, which gives \(x=9\). Step 6: The intersection points are therefore \((4, 4)\) and \((9, -6)\). Step 7: To find the area bounded by the curve and the line, integrate the difference between the curve and the line from \(y=-6\) to \(y=4\). Step 8: The area is given by the integral \(\int_{-6}^{4}\left(\frac{12-y}{2}-\frac{y^{2}}{4}\right)dy\). Step 9: Evaluate the integral to find the area. The calculation is not shown here, but the final result is \(41.67\). Step 10: The correct answer is \(41.67\), which corresponds to option C.Click to rate:87.6(941 votes)Search questionBy textBy image/screenshotDrop your file here orClick Hereto upload

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