Calculate the area under the curve y = 2x + 1 from x = 1 to x = 4.

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Question: Calculate the area under the curve y = 2x + 1 from x = 1 to x = 4.

Options:

Correct Answer: 15

Solution:

The area under the curve is given by ∫(from 1 to 4) (2x + 1) dx = [x^2 + x] from 1 to 4 = (16 + 4) - (1 + 1) = 20.

Calculate the area under the curve y = 2x + 1 from x = 1 to x = 4.

Calculate the area under the curve y = 2x + 1 from x = 1 to x = 4.

Step 1: Identify the function you want to find the area under. In this case, the function is y = 2x + 1.
Step 2: Set up the integral to calculate the area under the curve from x = 1 to x = 4. This is written as ∫(from 1 to 4) (2x + 1) dx.
Step 3: Find the antiderivative (the integral) of the function 2x + 1. The antiderivative is x^2 + x.
Step 4: Evaluate the antiderivative at the upper limit (x = 4). Calculate (4^2 + 4) = 16 + 4 = 20.
Step 5: Evaluate the antiderivative at the lower limit (x = 1). Calculate (1^2 + 1) = 1 + 1 = 2.
Step 6: Subtract the value from the lower limit from the value at the upper limit. This gives you 20 - 2 = 18.
Step 7: The area under the curve from x = 1 to x = 4 is 18.

Definite Integral – The process of calculating the area under a curve between two specified points using integration.
Fundamental Theorem of Calculus – Relates differentiation and integration, allowing the evaluation of definite integrals using antiderivatives.