Jim Lmbers MAT 8 Spring Semester 9- Leture Notes These notes orrespond to Setion. in Stewrt nd Setion 5. in Mrsden nd Tromb. Double Integrls over etngles In single-vrible lulus, the definite integrl of funtion f(x) over n intervl [, b ws defined to be b f(x) dx lim f(x i )Δx, where Δx (b )/n, nd, for eh i, x i x i x i, where x i + iδx. The purpose of the definite integrl is to ompute the re of region with urved boundry, using the formul for the re of retngle. The summtion used to define the integrl is the sum of the res of n retngles, eh with width Δx, nd height f(x i ), for i,,..., n. By tking the limit s n, the number of retngles, tends to infinity, we obtin the sum of the res of infinitely mny retngles of infinitely smll width. We define the re of the region bounded by the lines x, y, x b, nd the urve y f(x), to be this limit, if it exists. Unfortuntely, it is too tedious to ompute definite integrls using this definition. However, if we define the funtion F (x) s the definite integrl then we hve F (x) x [ F x+h (x) lim f(s) ds h h i f(s) ds, x f(s) ds h x+h x f(s) ds. Intuitively, s h, this expression onverges to the re of retngle of width h nd height f(x), divided by the width, whih is simply the height, f(x). Tht is, F (x) f(x). This leds to the Fundmentl Theorem of Clulus, whih sttes tht b f(x) dx F (b) F (), where F is n ntiderivtive of f; tht is, F f. Therefore, definite integrls re typilly evluted by ttempting to undo the differentition proess to find n ntiderivtive of the integrnd f(x), nd then evluting this ntiderivtive t nd b, the limits of the integrl. Now, let f(x, y) be funtion of two vribles. We onsider the problem of omputing the volume of the solid in 3-D spe bounded by the surfe z f(x, y), nd the plnes x, x b,
y, y d, nd z, where, b, nd d re onstnts. As before, we divide the intervl [, b into n subintervls of width Δx (b )/n, nd we similrly divide the intervl [, d into m subintervls of width Δy (d )/m. For onveniene, we lso define x i + iδx, nd y j + jδy. Then, we n pproximte the volume V of this solid by the sum of the volumes of mn boxes. The bse of eh box is retngle with dimensions Δx nd Δy, nd the height is given by f(x i, y j ), where, for eh i nd j, x i x i x i nd y j yj y j. Tht is, V f(x i, yj ) Δy Δx. i j We then obtin the ext volume of this solid by letting the number of subintervls, n, tend to infinity. The result is the double integrl of f(x, y) over the retngle {(x, y) x b, y d}, whih is lso written s [, b [, d. The double integrl is defined to be V f(x, y) da lim m, i j f(x i, yj ) Δy Δx, whih is equl to the volume of the given solid. The da orresponds to the quntity ΔA ΔxΔy, nd emphsizes the ft tht the integrl is defined to be the limit of the sum of volumes of boxes, eh with bse of re ΔA. To evlute double integrls of this form, we n proeed s in the single-vrible se, by noting tht if f(x, y), funtion of y, is integrble on [, d for eh x [, b, then we hve f(x, y) da lim m, lim lim i j lim i i b d f(x i, yj ) Δy Δx m j f(x i, yj )Δy Δx [ d f(x i, y) dy Δx Similrly, if f(x, y ), funtion of x, is integrble on [, b for eh y [, d, we lso hve f(x, y) da d b
This result is known s Fubini s Theorem, whih sttes tht double integrl of funtion f(x, y) n be evluted s two iterted single integrls, provided tht f is integrble s funtion of either vrible when the other vrible is held fixed. This is gurnteed if, for instne, f(x, y) is ontinuous on the entire retngle. Tht is, we n evlute double integrl by performing prtil integrtion with respet to either vrible, x or y, whih entils pplying the Fundmentl Theorem of Clulus to integrte f(x, y) with respet to only tht vrible, while treting the other vrible s onstnt. The result will be funtion of only the other vrible, to whih the Fundmentl Theorem of Clulus n be pplied seond time to omplete the evlution of the double integrl. Exmple Let [, [,, nd let f(x, y) x y + xy 3. We will use Fubini s Theorem to evlute We hve f(x, y) dy dx 8 3. x y + xy 3 dy dx x y + xy 3 dy dx x y dy + [x y dy + x [ x y + x y4 4 dx ( x 3 x + 4x dx 3 + x ) xy 3 dy dx y 3 dy dx In view of Fubini s Theorem, double integrl is often written s f(x, y) da f(x, y) dy dx f(x, y) dx dy. 3
Exmple We wish to ompute the volume V of the solid bounded by the plnes x, x 4, y, y, z, nd x + y + z 8. The plne tht defines the top of this solid is lso the grph of the funtion z f(x, y) 8 x y. It follows tht the volume of the solid is given by the double integrl V 8 x y da, [, 4 [,. Using Fubini s Theorem, we obtin V 8 x y da 8 x y dy dx (8y xy y 4 x dx (4x x ) 4 (56 6) (4 ) 7. ) dx We onlude by noting some useful properties of the double integrl, tht re diret generliztions of orresponding properties for single integrls: Linerity: If f(x, y) nd g(x, y) re both integrble over, then [f(x, y) + g(x, y) da f(x, y) da + Homogeneity: If is onstnt, then f(x, y) da Monotoniity: If f(x, y) on, then f(x, y) da. f(x, y) da g(x, y) da Additivity: If nd re disjoint retngles nd Q is retngle, then f(x, y) da f(x, y) da + f(x, y) da. Q 4
Prtie Problems. Evlute the following double integrls. () (b) () e x os y da, [, [, π/ e x+y, [, [, ln os(x + y), [, π/4 [, π/4 Hint: try using trigonometri identity to mke it esier to pply Fubini s Theorem.. Compute the volumes of the following solids bounded by the indited surfes. () x, x 3, y, y 4, z, x + y + z 9 (b) x, x, y, y, z, z y os ( ) πx Hint: use the properties of integrls on the previous pge. () x, y, y 4, z, z 4 x Additionl Prtie Problems Additionl prtie problems from the reommended textbooks re: Stewrt: Setion., Exerises -33 odd Mrsden/Tromb: Setion 5., Exerises, 3, 7 5