2. Evaluate the integral \( \int_{0}^{1} \int_{0}^{z} \int_{0}^{1-y}(x y z) d x d y d z \)

Suppose that the profit (in dollars) from the production and sale of \( x \) units of a product is given by \[ P(x)=180 x-\frac{x^{2}}{100}-400 \text {. } \] In addition, suppose that for a certain month the number of units produced on day \( t \) of the month is \[ x=q(t)=1000+10 t \text {. } \] (a) Find \( (P \circ q)(t) \) to express the profit (in dollars) as a function of the day of the month. ( \( P \circ q)(t)=\square \) (b) Find the number of units produced, and the profit, on the nineteenth day of the month. units produced profit

\( \int _{}^{}3\sqrt {x}+(\frac{10}{x^{6}}) d \)

Calcula el volumen del solido de revolución que forman las siguientes funciones: - Sea \( f: \mathbb{R} \rightarrow \mathbb{R} \) con regla de correspondencia \( f(x)=\frac{x^{2}}{4} \), considera el intervalo \( \quad[1,5] \) - \( \quad \) Sea \( g: \mathbb{R}_{+} \cup\{0\} \rightarrow \mathbb{R}_{+} \cup\{0\} \) con regla de correspondencia \( g(x)=2 \sqrt{x}+1 \), considera el intervalo \( [3,10] \) - Sea \( h: \mathbb{R} \rightarrow[-1,1] \) con regla de correspondencia \( h(x)=\operatorname{sen}(x) \), considera el intervalo \( [-\pi, \pi] \)

Deriva lo siguiente \( y=\sqrt[3]{2 x^{2}+3 x} \)

Approximate the definite integral using the Trapezoldal Rule with \( n=4 \). Compare the result with the approximation of the integral using a graphing utility. (Round your answers to three decimal places.) \[ \int_{0}^{2} x \ln (x+7) d x \] Trapezoidal graphing utility 4.276