How do I find the area bounded by the curve y = x^1/2 + 2, the x-axis, and the lines x = 1 and x = 4
see post just above
To find the area bounded by the curve, the x-axis, and the given lines, we can use the definite integral. Here are the steps to find the area:
1. Determine the limits of integration: In this case, the limits of integration are the x-values where the curve intersects the lines x=1 and x=4. Let's solve for these x-values:
When x = 1:
y = 1^1/2 + 2 = 1 + 2 = 3
So, we have one limit of integration as x=1.
When x = 4:
y = 4^1/2 + 2 = 2 + 2 = 4
So, we have the other limit of integration as x=4.
2. Set up the integral: The formula for finding the area between a curve and the x-axis is:
Area = ∫[a,b] (f(x) - g(x)) dx
In this case, f(x) is the given curve y = x^1/2 + 2, and g(x) is the x-axis, which is y = 0. So, the integral becomes:
Area = ∫[1,4] (x^1/2 + 2 - 0) dx
3. Simplify the integral: Expand the expression inside the integral and integrate term by term. Remember that the integral of x^k is (1/(k+1)) * x^(k+1).
Area = ∫[1,4] (x^1/2 + 2) dx
= ∫[1,4] x^1/2 dx + ∫[1,4] 2 dx
= (2/3) * x^(1/2) + 2x evaluated from 1 to 4 + 2 * (x evaluated from 1 to 4)
4. Evaluate the integral: Substitute the upper and lower limits of integration into the integral expression and calculate the result.
Area = [(2/3) * 4^(1/2) + 2 * 4] - [(2/3) * 1^(1/2) + 2 * 1]
= [(2/3) * 2 + 8] - [(2/3) * 1 + 2]
= [(4/3 + 8)] - [(2/3 + 2)]
= [20/3] - [8/3]
= 12/3
= 4 square units
Therefore, the area bounded by the curve y = x^1/2 + 2, the x-axis, and the lines x = 1 and x = 4 is 4 square units.