I'm getting the wrong answer for this question where I need to determine work. I think it's because I'm not sure how to figure in the weight of the bucket properly. Anyway, my answer was $2,681.28$ joules, but the correct answer should be $3,857$ joules.
What did I do wrong?
A leaky 10-kg bucket is lifted from the ground to a height of 12 m at a constant speed with a rope that weighs 0.8 kg/m. Initially the bucket contains 36 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. How much work is done?
Evaluate Work for Rope
$0.8 \cdot 9.8=7.84$ newtons per meter
$\int_0^{12}7.84xdx=564.48$ joules
Evaluate Work for Bucket
$(10+36)9.8=450.8$ newtons initially
$3\cdot 9.8=29.4$ newtons leaked/lost per meter
$\int^{12}_029.4xdx=2116.8$ joules
Combined joules
$564.48+2116.8=2681.28$ joules total
When integrating the work done on the bucket, you're integrating the weight of the water lost, not the weight of the bucket and the water retained (i.e. you're not integrating the weight that is actually lifted). It should be $$ \int_0^{12}(450.8 - 29.4x)dx = 3292.8 $$ When added to your $564.48J$ for the rope, this becomes a total of $3857.28J$, which is the answer you were after.