Find the area bounded by the curve y=1/2+2, the x-axis, and the lines x=1 and x=4.
a.7 1/2
b.10 2/3
c.16
d.28 1/2
Looks like (a) to me.
Assuming no typo in the equation.
integral of 1/2 +2 limits are 1 to 4
answer is A
To find the area bounded by the given curves, we need to integrate the function that represents the curve with respect to x, between the limits of integration.
In this case, we need to find the area bounded by the curve y = 1/2x + 2, the x-axis, and the lines x = 1 and x = 4.
First, let's find the x-values where the curve intersects the x-axis. Setting y = 0, we have:
0 = 1/2x + 2
Rearranging the equation:
1/2x = -2
Multiplying both sides by 2:
x = -4
So the curve intersects the x-axis at x = -4.
Next, let's find the x-values where the curve intersects the lines x = 1 and x = 4.
Setting x = 1, we have:
y = 1/2(1) + 2
y = 1/2 + 2
y = 2.5
So the curve intersects the line x = 1 at the point (1, 2.5).
Setting x = 4, we have:
y = 1/2(4) + 2
y = 2 + 2
y = 4
So the curve intersects the line x = 4 at the point (4, 4).
Now we can proceed to find the area bounded by the curve. We will split the area into two parts: the area above the curve and the area below the curve.
1. Area above the curve:
To find the area above the curve, we need to integrate the function y = 1/2x + 2 from x = 1 to x = 4.
∫[1 to 4] (1/2x + 2) dx
Integrating, we get:
[1/4x^2 + 2x] from 1 to 4
Substituting the limits of integration:
[(1/4(4)^2 + 2(4)] - [(1/4(1)^2 + 2(1)]
Simplifying:
[(1/4)(16) + 8] - [(1/4)(1) + 2]
= (4 + 8) - (1/4 + 2)
= 12 - (1/4 + 2)
= 12 - (1/4 + 8/4)
= 12 - 9/4
= 48/4 - 9/4
= 39/4
2. Area below the curve:
The area below the curve is a triangle with base length (4 - 1) = 3 and height (2.5 - 0) = 2.5.
Therefore, the area below the curve is (1/2)(3)(2.5) = 3.75.
The total area bounded by the curve is the sum of the area above and below the curve:
Area = Area above the curve + Area below the curve
= 39/4 + 3.75
To simplify, we convert 39/4 to a decimal by dividing 39 by 4, which equals 9.75.
Therefore, the total area bounded by the curve is:
Area = 9.75 + 3.75
= 13.5
So, the correct option is not among the given choices. The area bounded by the curve is 13.5 square units.