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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Definite Integration

NDA 2024 · Mathematics · Definite Integration

Let (a)=∫_a^ a+100 π | x| d x What is (a) equal to?

Let φ(a)=∫aa+100π∣sin⁡x∣dx\varphi(a)=\int_a^{a+100 \pi}|\sin x| d x What is φ(a)\varphi(a) equal to?

Options

  1. A0
  2. Baa
  3. C100a100 a
  4. D200

Correct answer

D. 200

Step-by-step solution

ϕ(a)=∫aa+100π∣sin⁡x∣dx\phi(a)=\int_a^{a+100 \pi}|\sin x| d x As period of ∣sin⁡x∣=πand∣sin⁡x∣=sin⁡xin|\sin x|=\pi and |\sin x|=\sin x inϕ(a)=∫0100π∣sin⁡x∣dx=100∫0π∣sin⁡x∣dx=100∫0πsin⁡xdxϕ(a)=100(−cos⁡x)0π=100×2=200\begin{aligned} \phi(a) & =\int_0^{100 \pi}|\sin x| d x \\ & =100 \int_0^\pi|\sin x| d x \\ & =100 \int_0^\pi \sin x d x \\ \phi(a) & =100(-\cos x)_0^\pi=100 \times 2=200 \end{aligned}$$
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