Different approach in potential energy

Question

The question is: In a region potential energy of a body is given as $$U = 4x^2 + 3x -5J $$ Find work done by external agent in slowly moving the body from point (3,2) to (-1,4) . The answer can be simply provided by subtracting the initial potential energy from final since the body is moved slowly. The final potential energy is -4j. Initial potential energy is 40j. So, work done by external agent will be $$ U_f - U_i = -4 -40 = -44J$$ I am new to integration and I wonder if this question can be done through integration by taking the sum of all small changes in potential energy. If it can be done how will one start making equation to integrate in the first step?

Answer 1

The "small change in potential energy" is its differential element $dU$

Differentiating the equation: $$dU = (8x+3)dx$$

Integrating this: $$\int_{x=3}^{x=-1}dU=\int_{x=3}^{x=-1}(8x+3)dx$$ $$[U]_{x=3}^{x=-1} = [4x^2+3x-5J]_{x=3}^{x=-1}$$ $$U_f-U_i=[4(-1)^2+3(-1)-5J]-[4(3)^2+3(3)-5J]$$ $$U_f-U_i=-4-40=-44J$$

The answer will of course we the same, but the integration was clearly unnecessary. Calculus is useful when we care about the instantaneous rates of changes in variables, not for questions like this.

As a side-note, the differential $-\frac{dU}{dx}$ is physically significant, as it is equal to the magnitude of force $F$. The integral $-\int Fdx$ is the work done on the body, giving us the change in potential energy. Maybe this is why you were interested in using integration?