Math 408, Actuarial Statistics I A.J. HildebrandDouble integralsTips on doing double integrals• Setting up double integrals: This means writing the integral over a given region (usuallydescribed verbally) as an iterated integral of the formR∗∗R∗∗f(x, y)dxdy and/orR∗∗R∗∗f(x, y)dydxwith specific limits in place of the asterisks. This is the part that usually causes the greatestdifficulties. Proceed as follows:– Sketch the given region. The most important piece of advice here. Don’t try to workout limits “in the abstract” or by algebraic manipulations; if you do so, you risk makingmistakes that could have been avoided or spotted with a simple sketch.– Try to “sweep” out the region either vertically or horizontally, and determine the upperand lower bounds on x and y that correspond to this sweep.– Using the bounds on x and y obtained from such a sweep, express the region in termsof inequalities of the form ∗ ≤ x ≤ ∗, ∗ ≤ y ≤ ∗, making sure that (at least) one of thevariables has constant limits. A horizontal sweep corresponds to constant limits on y,while a vertical sweeps corresponds to constant limits on x. In most cases, either typeof sweep will work, though often one is more convenient or easier to work with than theother.– Finally, use the bounds in these inequalities to set up a double integral over the givenregion R, i.e., express a double integralRRRf(x, y)dxdy as an iterated integral in the formR∗∗R∗∗f(x, y)dxdy and/orR∗∗R∗∗f(x, y)dydx with specific limits in place of the asterisks.– In some (rare) cases, you cannot get away with a single such iterated integral, and youhave to split R into two (or more) pieces, with each represented in the above form.• Note on the order of integration: The key rule here is that the outside integral musthave constant limits; an expression likeR2xxR10f(x, y)dxdy does not make sense. Once youhave determined bounds on x and y from a vertical or horizontal sweep of the region as above,the order is usually determined since normally only one of the variables will have constantlimits, and that variable therefore has to be the one on the outside integral.• Advice on notation: I strongly rec omme nd writing the integration variable involved explic-itly under each integral sign, using notations likeR1x=0R2xy =xxydydx,x2/2y 2xy =x, etc. Withoutsuch notational reminders it is easy to forget which of x and y is considered a variable andwhich is considered a constant in an integration, resulting in mistakes.• Some useful formulas: The following are some frequently occurring integrals. They areeasy to e valuate directly, but knowing these formulas saves valuable time in an exam setting.(There are some mild restrictions on the constant c here: In the first and second formulas, therestrictions on c are c 6= −1 respectively c 6= 1, in order for the fraction on the right to makesense. The third formula holds for c > 0, since otherwise the integral would be infinite.)Z10xcdx =1c + 1,Z∞11xcdx =1c − 1,Z∞0e−cxdx =1c,Z∞Le−cxdx =e−cLc(c > 0)1Math 408, Actuarial Statistics I A.J. HildebrandPractice problems on double integralsThe problems below illustrate the kind of double integrals that arise in actuarial exam problems, andmost are derived from past actuarial exam problems. The first group of questions asks to set up adouble integral of a general function f(x, y) over a giving re gion in the xy-plane. This means writingthe integral as an iterated integral of the formR∗∗R∗∗f(x, y)dxdy and/orR∗∗R∗∗f(x, y)dydx, withspecific limits in place of the asterisks. To do this, follow the steps above (most importantly, sketchthe given region). The remaining questions are evaluations of integrals over concrete functions.1. Set up a double integral of f (x, y) over the region given by 0 < x < 1, x < y < x + 1.2. Set up a double integral of f(x, y) over the part of the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, onwhich y ≤ x/2.3. Set up a double integral of f (x, y) over the part of the unit square on which x + y > 0.5.4. Set up a double integral of f (x, y) over the part of the unit square on which both x and y aregreater than 0.5.5. Set up a double integral of f(x, y) over the part of the unit square on which at least one ofx and y is greater than 0.5.6. Set up a double integral of f(x, y) over the part of the region given by 0 < x < 50 − y < 50on which both x and y are greater than 20.7. Set up a double integral of f(x, y) over the set of all points (x, y) in the first quadrant with|x − y| ≤ 1.8. EvaluateRRRe−x−ydxdy, where R is the region in the first quadrant in which x + y ≤ 1.9. EvaluateRRRe−x−2ydxdy, where R is the region in the first quadrant in which x ≤ y10. EvaluateRRR(x2+ y2)dxdy, where R is the region 0 ≤ x ≤ y ≤ L11. EvaluateRRRf(x, y)dxdy, where R is the region inside the unit square in which both coordi-nates x and y are greater than 0.5.12. EvaluateRRR(x−y +1)dxdy, where R is the region inside the unit s quare in which x+y ≥ 0.5.13. EvaluateR10R10x max(x,
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