I am so confused please help asap! (calc, integrals) please explain as well
Answers
Answer: A.) 8
Explanation:
Use u-substitution.
(1) Let u=x^3
By the power rule, du/dx=3x^2
Multiplying by dx and dividing by three, we have du/3=x^2dx
To find the new lower bound of integration, plug the old bound, -3, for x in equation (1). We get u=(-3)^3= -27
Similarly, when the upper bound 3 is plugged in, u=27
Now, replacing f(x^3) with f(u) and x^2dx with du/3:
[tex]\int\limits^{3}_{-3} {x^2f(x^3)} \, dx= \int\limits^{27}_{-27} \frac{f(u)}3 \, du \\=\frac{1}3\left[\int\limits^{0}_{-27} {f(u)} \, du+\int\limits^{27}_{0} {f(u)} \, du \right] (2)\\Observe:\int\limits^{0}_{-27} {f(u)} \, du=\int\limits^{27}_{0} {f(u)} du\; \text{ because f(x) is an even function}\\\text{Substitute the left hand side integral for the RHS in equation (2):}\\=\frac{1}3\left[2\int\limits^{27}_0 {f(u)} du\right]\\=\frac{1}3 (2)(12)=8[/tex]
since the value of the first integral of the question = 12, which is given. Although the variable is different than the given (u instead of x), it's still the same integral
Related Questions
Which of the following factors affect the productivity of a forest?. A) soil fertility B) water supply C) human activities D) local climate E) all of the above
Answers
Answer:
E. all of the above
Explanation:
The productivity of a forest is affected and defined by numerous factors. The soil fertility provides the nutrients for the growth of the plants, so the more fertile the soil is the more vegetation there will be, and vice versa. The water supply defines the type of vegetation in the forests, as well as how much that vegetation will produce. The higher amounts of water result in high productivity, while the smaller amounts of water in lower productivity. The climate is always a very important factor, as it can either have negative influence on the forest productivity if it is too harsh or be the basis for high productivity if it is more moderate. The human activities are the newest factor that influences the forest productivity. The humans, with their activities, are more often than not a serious threat to the forests, as most commonly it ends up with destruction of the forest.
Answer:
E) all of the above
hope this helps youu!!