Question Given \( P(x)=\frac{2 x+7}{15}, x=1,2,3 \); answer the following \( \quad 9 \) Numeric 10 points questions. What is \( \sum_{i-1}^{n} P\left(x_{i}\right) \) ? Type your answer...

\( \lim _ { n i n } \frac { ( n + 1 ) ^ { 3 } - ( n + 2 ) ^ { 2 } } { ( n - 2 ) ^ { 3 } - ( n + 3 ) ^ { 3 } } \)

3. Use the value of the first integral \( I \) to evaluate the two given integrals. \( I=\int_{0}^{3}\left(x^{3}-4 x\right) d x=\frac{9}{4} \) a. \( \int_{0}^{3}\left(24 x-6 x^{3}\right) d x \) a. \( \int_{0}^{3}\left(24 x-6 x^{3}\right) d x=\frac{-27}{2} \) (Simplify your answer.) \( \int_{3}^{0}\left(4 x-x^{3}\right) d x \) b. \( \int_{3}^{0}\left(4 x-x^{3}\right) d x=\square \frac{9}{4} \) (Simplify your answer.)

9/16/24, 1:33 PM 4. Use the definition of the definite integral to evaluate \( \int_{2}^{6}\left(x^{2}-4\right) d x \). \( \int_{2}^{6}\left(x^{2}-4\right) d x=\frac{160}{3} \) (Simplify your answer.)

6. Use the Fundamental Theorem of Calculus to evaluate the following definite integral. \( \int_{0}^{2} 15 e^{3 \mathrm{x}} \mathrm{dx} \) \( \int_{0}^{2} 15 e^{3 \mathrm{x}} \mathrm{dx}=5 e^{6}-5 \) (Type an exact answer.)

9/16/24, 1:33 PM 8. Use a change of variables to evaluate the definite integral. \( \int_{0}^{2} x \sqrt{4-x^{2}} d x \) \( \int_{0}^{2} x \sqrt{4-x^{2}} d x= \)