Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 4 Definite Integration Ex 4.1 Questions and Answers.

## Maharashtra State Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.1

I. Evaluate the following integrals as a limit of a sum.

Question 1.

\(\int_{1}^{3}(3 x-4) \cdot d x\)

Solution:

Let f(x) = 3x – 4, for 1 ≤ x ≤ 3

Divide the closed interval [1, 3] into n subintervals each of length h at the points

1, 1 + h, 1 + 2h, 1 + rh, ….., 1 + nh = 3

∴ nh = 2

∴ h = \(\frac{2}{n}\) and as n → ∞, h → 0

Here, a = 1

∴ f(a + rh) = f(1 + rh)

= 3(1 + rh) – 4

= 3rh – 1

Question 2.

\(\int_{0}^{4} x^{2} d x\)

Solution:

Let f(x) = x^{2}, for 0 ≤ x ≤ 4

Divide the closed interval [0, 4] into n subintervals each of length h at the points

0, 0 + h, 0 + 2h, ….., 0 + rh, ….., 0 + nh = 4

i.e. 0, h, 2h, ….., rh, ….., nh = 4

∴ h = \(\frac{4}{n}\) as n → ∞, h → 0

Here, a = 0

Question 3.

\(\int_{0}^{2} e^{x} d x\)

Solution:

Let f(x) = e^{x}, for 0 ≤ x ≤ 2

Divide the closed interval [0, 2] into n equal subntervals each of length h at the points

0, 0 + h, 0 + 2h, ….., 0 + rh, ….., 0 + nh = 2

i.e. 0, h, 2h, ….., rh, ….., nh = 2

∴ h = \(\frac{2}{n}\) and as n → ∞, h → 0

Here, a = 0

Question 4.

\(\int_{0}^{2}\left(3 x^{2}-1\right) d x\)

Solution:

Let f(x) = 3x^{2} – 1, for 0 ≤ x ≤ 2

Divide the closed interval [0, 2] into n subintervals each of length h at the points.

0, 0 + h, 0 + 2h, ….., 0 + rh, ……, 0 + nh = 2

i.e. 0, h, 2h, ….., rh, ….., nh = 2

∴ h = \(\frac{2}{n}\) and as n → ∞, h → 0

Here, a = 0

∴ f(a + rh) = f(0 + rh)

= f(rh)

= 3(rh)^{2} – 1

= 3r^{2}h^{2} – 1

Question 5.

\(\int_{1}^{3} x^{3} d x\)

Solution:

Let f(x) = x^{3}, for 1 ≤ x ≤ 3.

Divide the closed interval [1, 3] into n equal su bintervals each of length h at the points

1, 1 + h, 1 + 2h, ……, 1 + rh, ……, 1 + nh = 3

∴ nh = 2

∴ h = \(\frac{2}{n}\) and as n → ∞, h → 0

Here a = 1