Calculus intergrals

Question

If 𝑓(𝑥) is a positive continuous function on ℝ, what does the function 𝐹(𝑥)=∫𝑥𝑎𝑓(𝑡)𝑑𝑡 represent (for 𝑥>𝑎)?

The slope of the tangent line to the graph of 𝑓 at 𝑥.

The area between the graph of 𝑓(𝑥) and the 𝑦-axis between 𝑓(𝑎) and 𝑓(𝑥).

The area between the graph of 𝑓 on the 𝑥-axis on the interval [𝑎,𝑥].

The area of a rectangle with base on the horizontal axis.

Answer 1

It's the third choice. If we represent the integral by the limit of a Riemann sum, then each term in the sum corresponds to a narrow rectangle with its base on the x-axis and top at the graph. The sum of all those rectangles equals (approximately) the total area under the graph and above the x-axis, and, as the rectangles become infinitely narrow, the error in the approximation shrinks to zero, and the Riemann sum becomes the integral..