Volume of washers and disks

these are questions on volumes

Transcript

1. Kuta Software - Infinite Calculus Name___________________________________ Volumes of Revolution - Washers and Disks Date________________ Period____ For each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the x-axis. 2 2) y = 2 x + 2 2 y=x +2 1) y = − x + 1 y=0 y −8 −6 −4 y 8 8 6 6 4 4 2 2 −2 2 4 6 8 x −8 −6 −4 −2 2 −2 −2 −4 −4 −6 −6 −8 −8 4 6 8 x For each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the given axis. 4) x = − y 2 + 2 x= y Axis: x = −2 3) y = x + 1 2 y=x +1 Axis: y = −1 y y 8 8 6 6 4 4 2 2 −8 −6 −4 −2 2 4 6 8 x −8 −6 −4 −2 2 −2 4 6 8 x −2 −4 −4 −6 −6 −8 ©d 42j06185M bK0u6tgat rSXoUfltEw4abrpev wLSLbCn.b U kAKlilI BrfiIgnhDtqsf prMeAsBeWr1vleZda.6 G DMbabdsen Vwbigt2hu zIlnjf5ilnViVtGeM sC1aglMc8unlJuns3.j −8 -1- Worksheet by Kuta Software LLC
2. For each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the given axis. You may use the provided graph to sketch the curves and shade the enclosed region. 2 6) x = 5) y = x − 2 y = −2 x=2 Axis: y = −2 y+3 y +3 2 Axis: x = 1 x= y 8 y 8 6 6 4 4 2 2 −8 −6 −4 −2 2 4 6 8 x −8 −2 −6 −4 −2 2 4 6 8 x −2 −4 −4 −6 −6 −8 −8 Critical thinking questions: 7) Use the method of disks to derive the formula for the volume of a sphere of radius r. 8) A 6 cm diameter drill bit is used to drill a cylindrical hole through the middle of a sphere of radius 5 cm. What is the volume of the resulting object? ©B i2N0B1i5Z cKbu5tqa0 CSFobfQtywbaQryeu zLGLRCy.2 c rAal3lP QrViPg0hVtosu 5rEems9eKrev8exdw.q y YMjaadxeS lwLi8t4hG ZI6nbfcilnsi9t5eb gCwaclDc2uYlKuhsA.2 -2- Worksheet by Kuta Software LLC
3. Kuta Software - Infinite Calculus Name___________________________________ Volumes of Revolution - Washers and Disks Date________________ Period____ For each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the x-axis. 2 2) y = 2 x + 2 2 y=x +2 1) y = − x + 1 y=0 y −8 π −6 ∫ −4 y 8 8 6 6 4 4 2 2 −2 2 4 6 8 x −8 −6 −4 −2 2 −2 −2 −4 −4 −6 −6 −8 −8 1 4 6 8 x 2 (− x 2 + 1) 2 dx π −1 ∫ ((2x + 2) − (x + 2) ) dx 2 2 2 0 16 = π ≈ 3.351 15 = 48 π ≈ 30.159 5 For each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the given axis. 4) x = − y 2 + 2 x= y Axis: x = −2 3) y = x + 1 2 y=x +1 Axis: y = −1 y y 8 8 6 6 4 4 2 2 −8 −6 −4 −2 2 4 6 8 x −8 −6 −4 −2 2 4 6 8 x −2 −2 −4 −4 −6 −6 −8 −8 1 π ∫ (( x + 2) − (x + 2) ) dx 2 2 2 1 0 = π 49 π ≈ 5.131 30 ©l J2v0u1e5Z 0KhuOtfab ISeoxfbtJwOaGrEeS ALSLTC3.7 r VAVlLlK YrWizglhKtpsi prheSs6eOrfvYezdW.j 5 2MWaodYeW 2wlivt9hl rIznUfQiDnii0tleJ nC1aAl6cuuMlWu0sm.J ∫ ((− y + 4) − ( y + 2) ) dy 2 2 2 −2 = -1- 108 π ≈ 67.858 5 Worksheet by Kuta Software LLC
4. For each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the given axis. You may use the provided graph to sketch the curves and shade the enclosed region. 2 6) x = 5) y = x − 2 y = −2 x=2 Axis: y = −2 y+3 y +3 2 Axis: x = 1 x= y 8 y 8 6 6 4 4 2 2 −8 −6 −4 −2 2 4 6 8 x −8 −2 −6 −4 −2 2 4 6 8 x −2 −4 −4 −6 −6 −8 −8 π ∫ 2 ( x 2) 2 dx 4 0 = π 32 π ≈ 20.106 5 ∫( ( 2 ( )) y 2 y + 2) − +2 2 0 dy = 8π ≈ 25.133 Critical thinking questions: 7) Use the method of disks to derive the formula for the volume of a sphere of radius r. r r x2 + y2 = r2, y = r2 − x2 , V = π ∫ ( r2 − x2 ) −r 2 x3 dx, V = r 2 x − 3 ( ) 4 3 , V = πr 3 −r 8) A 6 cm diameter drill bit is used to drill a cylindrical hole through the middle of a sphere of radius 5 cm. What is the volume of the resulting object? 4 V =π 256π ( ( 25 − x ) − 9) dx = cm ∫ 2 −4 2 3 3 Create your own worksheets like this one with Infinite Calculus. Free trial available at KutaSoftware.com ©Q g220K1D5L TKfuotzaa zS2owfXt6w5air4ew IL7LKCH.j D iAol4lQ ZrXiygLhEthsg VrKeosleTrCveeEdU.e s BM5asdweq 6wQiltDhp PI1nzfGi4nhi4tkeO ACgaclFc3u3lyuMsX.P -2- Worksheet by Kuta Software LLC