I'm trying to solve this problem in a Calculus textbook.
My final answer is 541.6 Ib-ft
However, the solutions page says that the answer is 550 Ib-ft
Is my solution incorrect?
If you're wondering, the textbook is Calculus with Analytic Geometry by George Simmions 2ed Section 7.7 problem 5 page 249
Let $w$ be the weight of the body, then $$\frac{dw}{dx}=k$$ where $k$ is a constant. When the bucket starts it weight is $w(0)=5+60=65$ lb, then, in the top $\frac13(60)=20$ lb of sand has leaked, so the weight of the bucket is $w(10)=5+40=45$ lb. It follows $$\int_{65}^{45}dw=\int_0^{10}kdx\qquad\implies\qquad-20=10k\qquad\implies\qquad k=-2$$
The weight of the bucket is given by $w(x)=65-2x$. Thus, the work is given by $$W=\int_0^{10}(65-2x)dx=650-10^2=550\quad\text{lb-ft.}$$