082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

452 Chapter 13 Integration
2 Evaluate the followingintegrals:
∫ 1
(a) t 2 +1dt
0
∫ 1 1
(b)
0 1+t 2 dt
∫ 2
(c) 3e 2x −2e 3x dx
1
∫ 2
(d) (x +1)(x +2)dx
1
∫ 2
(e) 2sin4t dt
0
∫ ( ) π t
(f) 4cos dt
0 2
3 Evaluate the followingintegrals:
∫ 1
(a) t 2 +0.5t−6dt
0
∫ 3 3
(b)
2 2x + 2x
3 dx
∫ 2
(c) e −2x −3e −x dx
1
∫ 2
(d) 3sin(4t − π) +5cos(3t + π/2)dt
0
∫ 1
(e) 2tantdt
0
∫ 1.5 2
(f)
0 1+x 2 − 1
√ dx
4−x 2
4 Calculate the areabetween f (t) = cost andthet axis
ast variesfrom
(a) 0 to π (b) 0 to π 4 2
3π
(c) toπ (d) 0toπ
4
5 Calculate the total area between f (t) = cos2t andthe
t axisast variesfrom
(a) 0 to π 4
(b)
π
4 to π 2
(c) 0 to π 2
6 Calculate the area enclosed by the curvesy =x 2 and
y=x.
7 Calculate the area enclosed by the graphs ofthe
functionsy =t 2 +5 andy = 6.
8 Evaluate the following definiteintegrals:
∫ 1.5 1
(a)
1 t + 1 e t + 1
sint dt
∫ 4 5
(b)
1 8+3x 2 dx
∫ 1
(c) sinhx dx
−1
∫ 1
(d) coshx dx
−1
9 Calculate the area betweeny = 2tant andthet axis
for −1 t 1.4.
10 Findthe areabetweeny = sint,y = cost andthey
axis, fort 0.
11 Thevelocity, v,ofaparticleisgiven by
v=(1+t) 2
Findthe distancetravelled bythe particlefromt = 1
tot = 4; thatis,evaluate ∫ 4
1 vdt.
12 Evaluate the area under the functionx = 1/t for
1t10.
13 Evaluate
∫ 2
(a) x 1.4 dx
3
∫ π
(c) sinxcosx dx
0
∫ 1
(b) (e t ) 2 dt
0
∫ 2
(d) sinh 2 x dx
1
Solutions
1 (a) 20 (b) 1.3863 (c) 2
(d) 2.3504 (e) 0.5 (f) −1.9800
(g) − 1 3
4
2 (a)
3
53
(d)
6
(h) 0 (i) 0.3995
(b)
π
4
(e) 0.5728 (f) 8
(c) −184.75
3 (a)− 65
12
(b) 2.275 (c) −0.639
(d) −0.9255 (e) 1.2313 (f) 1.1175
4 (a) 0.7071 (b) 1
5 (a)
(c) 0.7071 (d) 2
1
2
(b)
1
2
(c) 1