Generate Quiz 6A332F3BE8962

The right response is 24.

Finding the area under the curve f(x) = x^2 + 1 dx between x = 4 and x = 1 is the question.

We must assess the definite integral in order to accomplish this:

∫(1 to 4) (x^2 + 1) dx

Using the integration power rule:

∫(x^2 + 1) dx = [x^3/3 + x] + C

Between the limits x = 1 and x = 4, the integral is evaluated:

[x^3/3 + x]_1^4

Changing the upper and lower bounds to:

(4^3/3 + 4) – (1^3/3 + 1)

= (64/3 + 4) – (1/3 + 1)

= 21 + 3 = 24

As a result, 24 is the area under the curve connecting x = 4 and x = 1.

Answer:

C is the right response.

The midpoint between the focus and the focus projection onto the directrix is known as the vertex of a parabola.

The directrix in this instance is the line x = 2, and the focus is at (0, 0). The point (2, 0) is where the focus is projected onto the directrix.

Consequently, the following represents the halfway point between the focus and its projection:

((0 + 2)/2, (0 + 0)/2) = (1, 0)

As a result, the parabola’s vertex is (1, 0).