Areas and double integrals. (Sect. 15.3)

Transcription

1 Areas and double integrals. (Sect. 5.) Areas of a region on a plane. Average value of a function. More eamples of double integrals. Areas of a region on a plane The area of a closed, bounded region on a plane is given b A = d d. emark: To compute the area of a region we integrate the function f (, ) = on that region. The area of a region is computed as the volume of a -dimensional region with base and height equal to.

2 Areas of a region on a plane Find the area of = {(, ) :, ],, + ]}. Solution: We epress the region as an integral Tpe I, integrating first on vertical directions: = + A = + d d. = A = ( + ) d = ( + ) d = ( + ). A = = 8 A = 9. Areas of a region on a plane Find the area of = {(, ) :, ],, + ]} integrating first along horizontal directions. Solution: We epress the region as an integral Tpe II, integrating first on horizontal directions: = A = d d + d d. = = + = = A = d d + We must get the same result: A = 9/. d d.

3 Areas of a region on a plane Find the area of = {(, ) :, ],, + ]} integrating first along horizontal directions. Solution: ecall: A = A = d + d d + ( + ) d d d. ( A = /) ( + / ) + A = = 6. We conclude that A = 9. Areas and double integrals. (Sect. 5.) Areas of a region on a plane. Average value of a function. More eamples of double integrals.

4 Average value of a function eview: The average of a single variable function. The average of a function f : a, b] on the interval a, b], denoted b f, is given b f = b f () d. (b a) a f f() a b The average of a function f : on the region with area A(), denoted b f, is given b f = A() f (, ) d d. Average value of a function Find the average of f (, ) = on the region = {(, ) :, ],, ]}. Solution: The area of the rectangle is A() = 6. We onl need to compute I = f (, ) d d. I = d d = I = 9 ( ) ( ) d = I = 9. 9 d. Since f = I /A() = 9/6, we get f = /.

5 Areas and double integrals. (Sect. 5.) Areas of a region on a plane. Average value of a function. More eamples of double integrals. More eamples of double integrals Find the integral of ρ(, ) = + in the triangle with boundaries =, = and =. Solution: We need to compute M = ρ(, ) dd. emark: If ρ is the mass densit, then M is the total mass. = = = M = M = ( + ) d d = ( + ] d = ) ( + )] d. M =.

6 More eamples of double integrals Given the function ρ(, ) = +, the number M computed in the previous eample, and the triangle with boundaries =, = and =, find the numbers r = ρ(, ) d d, r = ρ(, ) d d. M M emark: r = r, r is the center of mass of the bod. Solution: ecall: M =. We need to compute r = M r = ( + ) dd = + ] d = ( ) + ( r =. )] d More eamples of double integrals Given the function ρ(, ) = +, the number M computed in the previous eample, and the triangle with boundaries =, = and =, find the numbers r = ρ(, ) d d, r = ρ(, ) d d. M M Solution: ecall: M = and r =. r = M r = ( + ) dd = + 8 ] d = r = ] + = ( ) ( + ( ) + 8 ( )] )] d 7 6 r = 7 8.

7 More eamples of double integrals The centroid of a region in the plane is the vector c given b c =, d d, where A() = d d. A() emark: The centroid of a region can be seen as the center of mass vector of that region in the case that the mass densit is constant. When the mass densit is constant, it cancels out from the numerator and denominator of the center of mass. More eamples of double integrals Find the centroid of the triangle inside =, = and =. Solution: The area of the triangle is A() = d d = d = A() =. Therefore, the centroid vector components are given b c = d d = ( d = ) c =. c = d d = ( ) d = d = ( ) so c =. We conclude, c =,.

8 More eamples of double integrals emark: The moment of inertia of an object is a measure of the resistance of the object to changes in its rotation along a particular ais of rotation. The moment of inertia about the -ais and the -ais of a region in the plane having mass densit ρ : are given b, respectivel, I = ρ(, ) d d, I = ρ(, ) d d. If M denotes the total mass of the region, then the radii of gration about the -ais and the -ais are given b = I /M = I /M. The moment of inertia of an object. Find the moment of inertia and the radius of gration about the -ais of the triangle with boundaries =, = and =, and mass densit ρ(, ) = +. Solution: The moment of inertia I is given b I = I = ( + ) d d = ( 5 d = ) 5 ( ) + ( I = 5. )] d Since the mass of the region is M = /, the radius of gration along the -ais is = I /M = 5, that is, = 5.