5 Steps to a 5 AP Calculus BC 2019
262 STEP 4. Review the Knowledge You Need to Score High Solution: (a) Sinces= ∫t 0 f(x)dx, thenv(t)=s′(t)=f(t). Thus,v(4)=−8 ft/ ...
Areas, Volumes, and Arc Lengths 263 The area under the curve usingnrectangles of equal length is approximately: ∑n i= 1 (area of ...
264 STEP 4. Review the Knowledge You Need to Score High Or, find the area of each rectangle: Area of RectI=(f(0))Δx 1 =(1) ( 1 2 ...
Areas, Volumes, and Arc Lengths 265 Or, using ∑ notation: ∑^5 i= 1 f(xi)Δxi= ∑^5 i= 1 f(4+i)(1)= ∑^5 i= 1 √ 4 + 1. Enter ∑(√ ( 4 ...
266 STEP 4. Review the Knowledge You Need to Score High Trapezoidal Approximations Another method of approximating the area unde ...
Areas, Volumes, and Arc Lengths 267 Area under the curve: ≈ π 4 · 1 2 [ cos(0)+2 cos ( π/ 4 2 ) +2 cos ( π/ 2 2 ) +2 cos ( 3 π/ ...
268 STEP 4. Review the Knowledge You Need to Score High Ifx=g(y) is continuous and non-negative on [c,d], then the area under th ...
Areas, Volumes, and Arc Lengths 269 Step 3. Evaluate the integrals. ∣∣ ∣∣ ∫ 1 0 (x−1)^3 dx ∣∣ ∣∣= ∣∣ ∣ ∣∣ (x−1)^4 4 ] 1 0 ∣∣ ∣ ∣ ...
270 STEP 4. Review the Knowledge You Need to Score High Step 3. Evaluate the integrals. ∫− 1 − 2 ( x^2 − 1 ) dx= x^3 3 −x ]− 1 − ...
Areas, Volumes, and Arc Lengths 271 Example 4 Using a calculator, find the area bounded byf(x)=x^3 +x^2 − 6 xand thex-axis. See ...
272 STEP 4. Review the Knowledge You Need to Score High ∫k 0 sinxdx= 2 ∫π k sinxdx −cosx]k 0 = 2 [−cosx]πk −cosk−(−cos(0))= 2 (− ...
Areas, Volumes, and Arc Lengths 273 Example 1 Find the area of the regions bounded by the graphs of f(x)=x^3 andg(x)=x. (See Fig ...
274 STEP 4. Review the Knowledge You Need to Score High Note: You can use the symmetry of the graphs and let area = 2 ∫ 1 0 ( x− ...
Areas, Volumes, and Arc Lengths 275 Example 3 Using a calculator, find the area of the region bounded byy=sinxandy= x 2 between ...
276 STEP 4. Review the Knowledge You Need to Score High Step 2. Set up an integral. Area = ∫e 3 e ( 1 x −(−5) ) dx. Step 3. Eval ...
Areas, Volumes, and Arc Lengths 277 Example 1 The base of a solid is the region enclosed by the ellipse x^2 4 + y^2 25 =1. The c ...
278 STEP 4. Review the Knowledge You Need to Score High Example 2 Find the volume of a pyramid whose base is a square with a sid ...
Areas, Volumes, and Arc Lengths 279 Example 3 The base of a solid is the region enclosed by a triangle whose vertices are (0, 0) ...
280 STEP 4. Review the Knowledge You Need to Score High The Disc Method The volume of a solid of revolution using discs: Revolvi ...
Areas, Volumes, and Arc Lengths 281 y x f(x) 0 k ab y = k h 0 g(y) y x x = h d c Figure 12.4-6 Example 1 Find the volume of the ...
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