I cant figure out how to set up the integrals.

1.If the work required to stretch a spring 3 ft beyond its natural length is 15 ft-lb, how much work is needed to stretch it 27 in. beyond its natural length?

2.A spring has a natural length of 12 cm. If a 28-N force is required to keep it stretched to a length of 20 cm, how much work W is required to stretch it from 12 cm to 16 cm? (Round your answer to two decimal places.)

F = kx --- use that to get k.

work = ∫F(x) dx

To set up the integrals for these problems, we need to understand the concept of work done against a force. Work is defined as the integral of the dot product between the force and the displacement vector. In simpler terms, work is the product of force and displacement.

In both of these problems, we are dealing with a spring, which exerts a force that is directly proportional to its displacement from its natural length.

1. Let's set up the integral for the first problem:
To stretch the spring 3 ft beyond its natural length, the work required is 15 ft-lb. We need to find the work needed to stretch it 27 inches beyond its natural length.
Let x represent the displacement in inches. So the force required to stretch the spring at any given displacement x is directly proportional to x. We can express this force as F(x) = kx, where k is the spring constant.
From the given information, we know that F(36) = 15, where 36 represents 3 ft in inches. We can use this information to find the value of k.
So, 15 = k * 36 => k = 15/36 = 5/12
The work done to stretch the spring 27 inches beyond its natural length can be calculated by integrating F(x) from 36 to 63 (27 inches beyond 36 inches):
W = ∫[36 to 63] (kx) dx
To calculate this integral, we substitute k = 5/12 and evaluate the integral.

2. Now let's set up the integral for the second problem:
To stretch the spring from 12 cm to 16 cm, we need to find the work required.
Similar to the first problem, the force required to stretch the spring at any given displacement x is directly proportional to x. We can express this force as F(x) = kx, where k is the spring constant.
From the given information, we know that F(20) = 28, where 20 represents 20 cm. We can use this information to find the value of k.
So, 28 = k * 20 => k = 28/20 = 7/5
The work done to stretch the spring from 12 cm to 16 cm can be calculated by integrating F(x) from 12 to 16:
W = ∫[12 to 16] (kx) dx
To calculate this integral, we substitute k = 7/5 and evaluate the integral.

By setting up the appropriate integrals and calculating them, we can find the work required to stretch the spring in both scenarios.