TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type

Students must practice these Maths 2B Important Questions TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type to help strengthen their preparations for exams.

TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type

Question 1.
Evaluate \(\int_0^{\pi / 2} \frac{a \sin x+b \cos x}{\sin x+\cos x} d x\). [(AP) May ’17]
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q1
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q1.1
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q1.2

Question 2.
Evaluate \(\int_{-\pi / 2}^{\pi / 2} \frac{\cos x}{1+e^x} d x\). [(TS) May ’17]
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q2
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q2.1

TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type

Question 3.
Evaluate \(\int_0^\pi \frac{1}{3+2 \cos x} d x\)
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q3
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q3.1

Question 4.
Evaluate \(\int_{-3}^3\left(9-x^2\right)^{3 / 2} d x\)
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q4
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q4.1

Question 5.
Evaluate \(\int_{-a}^a x^2\left(a^2-x^2\right)^{3 / 2} d x\). [(TS) May ’17]
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q5
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q5.1

Question 6.
Evaluate \(\int_0^4\left(16-x^2\right)^{5 / 2} d x\). [(AP) May ’19]
Solution:
Put x = 4 sin θ
dx = 4 cos θ dθ
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q6

Question 7.
Evaluate \(\int_0^2\left(x^2+1\right) d x\) as the limit of a sum.
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q7
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q7.1

TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type

Question 8.
Evaluate \({Lt}_{n \rightarrow \infty} \sum_{i=1}^n \frac{i^3}{i^4+n^4}\)
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q8
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q8.1

Question 9.
Evaluate \(\underset{n \rightarrow \infty}{L t}\left[\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right) \ldots\left(1+\frac{n}{n}\right)\right]^{1 / n}\)
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q9
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q9.1
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q9.2

Question 10.
Evaluate \({Lt}_{n \rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right) \cdots\left(1+\frac{n^2}{n^2}\right)\right]^{1 / n}\)
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q10
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q10.1
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q10.2
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q10.3

TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type

Question 11.
Find the area bounded between the curves y2 = 4x, y2 = 4(4 – x). [(TS) May ’19, ’11]
Solution:
Given curves are y2 = 4x
⇒ y = 2√x ………(1)
y2 = 4(4 – x) ………(2)
⇒ y = \(\sqrt{4(4-x)}\)
Solving (1) and (2)
4x = 4(4 – x)
⇒ x = 4 – x
⇒ 2x = 4
⇒ x = 2
from (1) ⇒ y = ± 2√2
Points of Intersections of (1) and (2) are A = (2, 2√2) and B = (2, -2√2)
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q11
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q11.1

Question 12.
Find the area enclosed between y = x2 – 5x and y = 4 – 2x.
Solution:
Given curves are
y = x2 – 5x ………(1)
y = 4 – 2x ………..(2)
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q12
Solving (1) and (2)
x2 – 5x = 4 – 2x
⇒ x2 – 3x – 4 = 0
⇒ (x – 4) (x + 1) = 0
∴ x = 4 and x = -1
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q12.1
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q12.2

Question 13.
Find the area enclosed between the curves y = 4x – x2, y = 5 – 2x. [(TS) Mar. ’16]
Solution:
Given curves are
y = 4x – x2 ………(1)
y = 5 – 2x ………(2)
Solving (1) and (2)
4x – x2 = 5 – 2x
⇒ x2 – 6x + 5 = 0
⇒ x2 – 5x – x + 5 = 0
⇒ x(x – 5) – 1(x – 5) = 0
⇒ x = 1 or 5
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q13
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q13.1

TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type

Question 14.
Find the area between the parabolas y2 = 4x and x2 = 4y. [(AP) Mar. ’20; (TS) ’17; May ’14]
Solution:
Given equations of curves are
y2 = 4x ………(1)
and x2 = 4y ……..(2)
Solving (1) and (2) the points of intersection can be obtained.
y2 = 4x
⇒ y4 = 16x2
⇒ y4 = 64y
⇒ y = 4
∴ 4x = y2
⇒ 4x = 16
⇒ x = 4
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q14
Points of intersection are (0, 0) and (4, 4).
∴ The area bounded between the parabolas
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L1 Q14.1

Question 15.
Evaluate \(\int_0^2 e^x d x\) as the limit of the sum.
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q1

Question 16.
Evaluate \(\int_0^4\left(x+e^{2 x}\right) d x\) as the limit of a sum.
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q2
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q2.1
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q2.2

Question 17.
Evaluate \(\int_0^{16} \frac{x^{1 / 4}}{1+x^{1 / 2}} d x\)
Solution:
L.C.M of 2, 4 is 4
Put x = t4
⇒ dx = 4t3 dt
L.L: x = 0 ⇒ t = 0
U.L: x = 16 ⇒ t = 2
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q3
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q3.1

Question 18.
Evaluate \(\int_0^{\pi / 4} \log (1+\tan x) d x\). [(AP) Mar. ’19, ’16; May ’18, (TS) ’16]
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q4
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q4.1

Question 19.
Evaluate \(\underset{n \rightarrow \infty}{L t} \frac{1}{n}\left[\tan \frac{\pi}{4 n}+\tan \frac{2 \pi}{4 n}+\ldots \ldots+\tan \frac{n \pi}{2 n}\right]\)
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q5

TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type

Question 20.
Find \({Lt}_{n \rightarrow \infty}\left(\frac{n !}{n^n}\right)^{1 / n}\)
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q6
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q6.1

Question 21.
Evaluate \(\int_0^5 x^3\left(25-x^2\right)^{7 / 2} d x\)
Solution:
Put x = 5 sin θ, then dx = 5 cos θ dθ
L.L: x = 0 ⇒ θ = 0
U.L: x = 5 ⇒ θ = \(\frac{\pi}{2}\)
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q7

Question 22.
Evaluate \(\int_0^2 x \sqrt{2-x} d x\)
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q8

Question 23.
Find \(\int_0^1 x^{3 / 2} \sqrt{1-x} d x\)
Solution:
Put x = sin2θ then dx = 2 sin θ cos θ dθ
Lower limit: x = 0 ⇒ θ = 0
Upper limit: x = 1 ⇒ θ = \(\frac{\pi}{2}\)
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q9

Question 24.
Evaluate \(\int_0^1 \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) d x\)
Solution:
Put x = tan θ, then θ = tan-1x
dx = sec2θ dθ
L.L: x = 0 ⇒ θ = 0
U.L: x = 1 ⇒ θ = \(\frac{\pi}{4}\)
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q10
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q10.1

TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type

Question 25.
Evaluate \(\int_0^1 x \tan ^{-1} x d x\)
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q11
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q11.1

Question 26.
Find the area enclosed by the curves y = 3x and y = 6x – x2.
Solution:
Given curves are
y = 3x ……..(1)
y = 6x – x2 ………(2)
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q12
Solving (1) and (2)
3x = 6x – x2
⇒ x2 = 3x
⇒ x(x – 3) = 0
⇒ x = 0 and x = 3
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q12.1

Question 27.
Find the area bounded between the curves y = 4x – x2, y = 5 – 2x.
Solution:
Given curves are
y = 4x – x2 ……..(1)
y = 5 – 2x ……..(2)
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q13
Solving (1) and (2)
4x – x2 = 5 – 2x
⇒ x2 – 6x + 5 = 0
⇒ (x – 1) (x – 5) = 0
⇒ x = 1, x = 5
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q13.1

Question 28.
Find the area bounded between the curves y = 2 – x, y = x2. [Mar. ’01]
Solution:
Given curves are
y = 2 – x2 ……(1)
y = x2 ………(2)
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q14
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q14.1

TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type

Question 29.
Find the area of the region enclosed by the curves y = sin x, y = cos x, x = 0, x = \(\frac{\pi}{2}\).
Solution:
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q15
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q15.1
TS Inter Second Year Maths 2B Definite Integrals Important Questions Short Answer Type L2 Q15.2

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