Let (a, b) be the point of intersection of the curve x2=2y and the straight line y−2x−6=0 in the second quadrant. Then the integral I=∫ab1+5x9x2dx is equal to:
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Step-by-Step Solution
Step 1: Find the intersection point (a, b)
First, we need to find the coordinates (a, b) where the curve x2=2y and the line y−2x−6=0 intersect. We can substitute the expression for y from the curve equation into the line equation to get a quadratic equation in terms of x.
Step 2: Solve the quadratic equation for x
Multiply the equation by 2 to clear the fraction, then rearrange it into standard quadratic form. Factor the quadratic equation to find the possible values for x.
Step 3: Determine (a, b) in the second quadrant
Substitute the x values back into y=2x2 to find the corresponding y values. The problem specifies that the intersection point (a, b) is in the second quadrant, where x<0 and y>0. Therefore, (a,b)=(−2,2).
Step 4: Set up the integral with limits
Now that we have found a=−2 and b=2, we can set up the definite integral with these limits.
Step 5: Use the property of definite integrals
We use the property of definite integrals ∫abf(x)dx=∫abf(a+b−x)dx. Here, a=−2 and b=2, so a+b−x=−2+2−x=−x. Applying this property transforms the integrand.
Step 6: Add the original and transformed integrals
Let the original integral be I. We have a second expression for I from the property. Adding these two expressions for I allows us to simplify the integrand significantly, as the denominators become common and the numerators combine to cancel out the 1+5x term.
Step 7: Evaluate the simplified integral
Now, we evaluate the simplified integral 2I=∫−229x2dx. We find the antiderivative of 9x2, which is 3x3, and evaluate it at the limits of integration. Finally, we divide by 2 to find the value of I.