Math Snap
PROBLEM
Calculate the work done by the force from to :
STEP 1
Assumptions1. The force acting on the object is given by the function .
. The object is moved along the x-axis from to .
3. The work done by the force is given by the definite integral of the force function from to .
STEP 2
We need to evaluate the definite integral of the force function from to . The integral is given by
STEP 3
We can split the integral into two separate integrals
STEP 4
Now, we need to find the antiderivative of each function.The antiderivative of is , and the antiderivative of is .
So, we have$$W = [\ln|x|]_{3}^{} + [2x^2]_{3}^{}
$$
STEP 5
Next, we need to evaluate each antiderivative at the upper and lower limits of the integral.
For the first term, we have$$[\ln|x|]_{3}^{5} = \ln|5| - \ln|3|
[2x^2]_{3}^{5} =2(5)^2 -2(3)^2$$
STEP 6
Now, we can simplify each term$$\ln|5| - \ln|3| = \ln\left(\frac{5}{3}\right)
2(5)^2 -2(3)^2 =2(25 -9) =2(16) =32$$
SOLUTION
Finally, we add the two terms together to find the total work doneSo, the work done by the force moving an object along the -axis from to is .