Multivariable and Vector Calculus · Multiple integration and change of variables
Let D be the closed region in the xy-plane bounded by the parabola y=x^2 and the line…
Problem
Let \(D\) be the closed region in the \(xy\)-plane bounded by the parabola \(y=x^{2}\) and the line \(y=x+2\). (a) Find the points of intersection of the two curves. Describe \(D\) as a Type I region (vertical strips) by giving a single interval of \(x\)-values together with lower and upper bounding functions of \(x\). (b) Evaluate \[ \iint_{D}(3x+2y)\,dA \] as an iterated integral in the order \(dy\,dx\). Give an exact value. (c) Describe \(D\) as a Type II region (horizontal strips). Split the \(y\)-range at the \(y\)-coordinate of the left-hand intersection point, and write the corresponding iterated integral (or sum of integrals) in the order \(dx\,dy\). Do not evaluate.
Hint
Intersection points come from equating the two bounding formulas. Vertical strips run from the parabola up to the line; horizontal strips change character once the line, rather than the left branch of the parabola, becomes the left wall.
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