Find the area between the x -axis and the graph y = x from x = 1 to x = 3.
4 square units
2 square units
I'm confused
6 square units
3 square units
did you draw the graph? It's just a trapezoid.
Its two vertical bases have height 1 and 3
The altitude is 3-1 = 2
So, the area is (1+3)/2 * 2 = 4
Oh Ok thanks, that makes sense!
To find the area between the x-axis and the graph y = x from x = 1 to x = 3, you need to integrate the function y = x with respect to x over the given interval.
First, let's find the definite integral of y = x from x = 1 to x = 3:
∫[1 to 3] x dx
To evaluate this integral, we can use the power rule of integration, which states that the integral of x to the power of n is equal to (x^(n+1))/(n+1), except when n equals -1 (ln(x)) or 0 (x). Since we have x to the power of 1, we can use this rule:
∫[1 to 3] x dx = (x^2)/2 [1 to 3]
Next, we substitute the upper and lower limits of integration:
[(3^2)/2 - (1^2)/2] = (9/2 - 1/2) = 8/2 = 4
Therefore, the area between the x-axis and the graph y = x from x = 1 to x = 3 is 4 square units.
So, the correct answer is 4 square units.