Applications of the Definite Integral ( Areas and Volumes)

Transcription

1 Mth1242 Project II Nme: Applictions of the Definite Integrl ( Ares nd Volumes) In this project, we explore some pplictions of the definite integrl. We use integrls to find the re etween the grphs of two functions nd the volume of some solids. I Are etween curves 1) We need to find the re of the region enclosed etween the grphs of the two functions f ( x) cos x nd g ( x) x 2 2. ) Using your clcultor in the Rdin Mode, Y 1 = cos(x), Y 2 = X 2-2, sketch oth grphs to the right, using the window [-5,5] [-5,5]. ) In the home screen, Press 2nd PRGM to get the DRAW menu, Choose Shde nd then get Y 1 nd Y 2 from VARS 1:Function to hve Shde(Y 2, Y 1 ) in your home screen. Here Y 2 is the lower function nd Y 1 is the upper function. Copy the shded region from the clcultor on your grph. c) To find the re of the shded region, we need to find first the points of intersections. Press 2nd TRACE, choose 5:Intersection, nswer the questions, nd get: The coordintes of the 1 st point of intersection (the one to the left) re ( , ) The coordintes of the 2 nd point of intersection (the one to the right) re (, ) To store these x-vlues in memory, type on your home screen, press STO, ALPHA, A, ENTER. Whenever you need the x-coordinte of the first point of intersection, you type A. Enter the x-coordinte of the second point of intersection s B. Using MATH 9, the re etween the 2 curves: B 2 Are = [cos x ( x 2)] dx = fnint(y 1 Y 2, X, A, B) = A d) Turn off the grphs of Y 1 nd Y 2 y pressing Y=, moving the cursor left to the highlighted = sign, nd pressing ENTER. Do the sme for Y 2. Now oth = signs re not highlighted nd the grphs re turned off. Grph the function Y 3 = cos(x) X on the intervl (A, B) y setting Xmin = A nd Xmx = B. Shde the re with Shde(0, Y 3 ). Copy your grph in the spce provided to the right. Is this region the sme s the shded region in prts ) nd )? e) Find the re of the region under the grph of Y 3. Is this re equl to the re in prt c)?

2 2) In ech of the following 4 cses, use your clcultor to sketch the grph of the functions, copy them on this pper, nd shde the re of the region(s) enclosed etween them. Then set up the integrl needed to find the enclosed re(s) nd use your clcultor to evlute it. ) f ( x) 3 nd g( x) for 0 x 1. ) f ( x) 3 nd g( x) for 1 x 1. c) f ( x) 3 nd g( x) (Note tht the region enclosed y these two curves consists of two prts. You need to find the points of intersection first) d) f ( x) 3, g( x), nd h( x) 9 x (Note tht the region enclosed y these three curves consists of only one prt. You need to find the points of intersection first)

3 II Volume of Solids: To find the volume V of solid, perform one of the following 2 methods: Verticl slicing, find the re A of the slice s function of x, nd integrte with respect to x: V A( x) dx Horizontl slicing, find the re A of the slice s function of y, nd integrte with respect to y: V A( y) dy 2 3) Consider the region R ounded y y 9 x, y = 0, nd x = 0 (in the first qudrnt). (Note: you my use MATH 9 nd your nswers should e correct to 4 deciml plces) ) Sketch the grph nd find the volume of the solid otined y rotting R out the x-xis. Verticl or Horizontl slicing? Slice is Disk or Wsher? Are of slice A = ) Sketch the grph nd find the volume of the solid otined y rotting R out the line y = 9. Verticl or Horizontl slicing? Slice is Disk or Wsher? Are of slice A =

4 c) Sketch the grph nd find the volume of the solid otined y rotting R out the line y = 1. Verticl or Horizontl slicing? Slice is Disk or Wsher? Are of slice A = d) Sketch the grph nd find the volume of the solid otined y rotting R out the y xis. Verticl or Horizontl slicing? Slice is Disk or Wsher? Are of slice A = e) Sketch the grph nd find the volume of the solid otined y rotting R out the line x = 3. Verticl or Horizontl slicing? Slice is Disk or Wsher? Are of slice A = f) Sketch the grph nd find the volume of the solid otined y rotting R out the line x = 4. Verticl or Horizontl slicing? Slice is Disk or Wsher? Are of slice A =

5 2 4) Consider the region R ounded y the curves y x nd y x. Sketch the grph, set up 2 formuls to find the volume of the solid otined y rotting R (Slicing nd Cylindricl shells), nd evlute these integrls using your clcultor: ) Aout the x_xis. ) Aout the y_xis. c) Aout the line x = 1.

6 d) Aout the line x = 2. e) Aout the line x = -2. f) Aout the line y = 1. g) Aout the line y = -1.