Calculate the area under the curve y = cos(x) from x = 0 to x = π/2.

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Question: Calculate the area under the curve y = cos(x) from x = 0 to x = π/2.

Options:

Correct Answer: 1

Solution:

The area under the curve y = cos(x) from x = 0 to x = π/2 is given by ∫(from 0 to π/2) cos(x) dx = [sin(x)] from 0 to π/2 = 1 - 0 = 1.

Calculate the area under the curve y = cos(x) from x = 0 to x = π/2.

Calculate the area under the curve y = cos(x) from x = 0 to x = π/2.

Correct Answer: 1

Step 1: Identify the function you want to find the area under, which is y = cos(x).
Step 2: Determine the limits of integration, which are from x = 0 to x = π/2.
Step 3: Set up the integral to calculate the area: ∫(from 0 to π/2) cos(x) dx.
Step 4: Find the antiderivative of cos(x), which is sin(x).
Step 5: Evaluate the antiderivative at the upper limit (π/2) and the lower limit (0): sin(π/2) - sin(0).
Step 6: Calculate sin(π/2), which equals 1, and sin(0), which equals 0.
Step 7: Subtract the two results: 1 - 0 = 1.
Step 8: Conclude that the area under the curve from x = 0 to x = π/2 is 1.

Definite Integral – The process of calculating the area under a curve by evaluating the integral of a function over a specified interval.
Trigonometric Functions – Understanding the properties and behavior of trigonometric functions, specifically the cosine function in this case.
Fundamental Theorem of Calculus – The theorem that connects differentiation and integration, allowing the evaluation of definite integrals using antiderivatives.