Question

Find the total area between the function f(x)=2x and the x-axis over the interval [−3,3].

Answer 1

The total area between the function f(x) = 2x and the x-axis over the interval [-3,3] is 0 square units.

Step-by-step explanation:

To find the total area between the function f(x) = 2x and the x-axis over the interval [-3,3], we can integrate the function from -3 to 3. The area under a curve can be calculated by finding the definite integral of the function. In this case, the integral of f(x) = 2x is:

∫(2x) dx from -3 to 3

Integrating 2x with respect to x gives us x^2. Evaluating the integral from -3 to 3 gives us:

x^2 evaluated from -3 to 3 = (3)^2 - (-3)^2 = 9 - 9 = 0

Therefore, the total area between the function f(x) = 2x and the x-axis over the interval [-3,3] is 0 square units.

Answer 2

The total area between the linear function f(x)=2x and the x-axis over the interval [−3,3] is 18 square units, calculated by finding the area of the triangle formed from 0 to 3 and doubling it due to symmetry.

Step-by-step explanation:

To find the total area between the function f(x)=2x and the x-axis over the interval [−3,3], we need to consider the symmetry of the function and calculate the area above and below the x-axis separately, since the function is linear and crosses the x-axis at the origin (0,0).

On the interval [−3,0], the function is negative, and on the interval [0,3], it is positive. As f(x)=2x is an odd function, the absolute value of the area from −3 to 0 is the same as the area from 0 to 3. So, we only need to calculate the area for one side and then double it.

The area of the triangle formed by the portion of the graph from 0 to 3 is given by A = (base × height) / 2 = (3 × (2×3)) / 2 = 9 square units. Since the function is odd, the total area is twice this value, which gives us 18 square units.