Oil production An oil refinery produces oil at a variable...

Oil production An oil refinery produces oil at a variable rate given by $$Q^{\prime}(t)=\left\{\begin{array}{ll} 800 & \text { if } 0 \leq t<30 \\ 2600-60 t & \text { if } 30 \leq t<40 \\ 200 & \text { if } t \geq 40 \end{array}\right.$$, where $t$ is measured in days and $Q$ is measured in barrels. a. How many barrels are produced in the first 35 days? b. How many barrels are produced in the first 50 days? c. Without using integration, determine the number of barrels produced over the interval [60,80]

Oil production An oil refinery produces oil at a variable rate given by $$Q^{\prime}(t)=\left\{\begin{array}{ll}
800 & \text { if } 0 \leq t<30 \\
2600-60 t & \text { if } 30 \leq t<40 \\
200 & \text { if } t \geq 40
\end{array}\right.$$, where $t$ is measured in days and $Q$ is measured in barrels.
a. How many barrels are produced in the first 35 days?
b. How many barrels are produced in the first 50 days?
c. Without using integration, determine the number of barrels produced over the interval [60,80]

Key Concepts

Definite Integration

Definite integration is used to calculate the accumulation of a quantity over an interval when given its rate of change. In contexts such as production rates or movement, integrating a rate function over a given time period yields the total quantity produced or distance traveled, making it a fundamental tool in calculus for linking rates to accumulations.

Piecewise?Defined Functions

Piecewise?defined functions specify different expressions for different intervals of the independent variable, requiring separate attention for each interval when performing calculations like integration or differentiation. They are essential in modeling situations where the behavior of a system changes according to conditions or time intervals.

Accumulation Functions

Accumulation functions are constructed by integrating a rate function over time to determine the total accumulation of a quantity. This concept is crucial in understanding how continuous changes contribute to an overall outcome, such as total production over a given period.

Area Under a Curve

The area under a curve in a graphical representation of a rate function corresponds to the accumulated quantity over an interval. This visualization helps in understanding how integration sums the contributions of a rate function across time, whether the rate is constant or variable.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration. It states that if a function is continuous over an interval, then its integral over that interval can be computed using its antiderivative evaluated at the boundaries of the interval, thereby linking the instantaneous rate of change to total accumulation.

Transcript

00:05 Okay, so we're going to start with this.

00:07 We have a oil production where q is our oil production, and q is measured in barrels.

00:16 So what we're going to do, q prime of t represents how many barrels we're making.

00:21 So within the first 30 days, tea is in days.

00:28 Within the first 30 days, we're making 800 barrels of oil a day.

00:34 So we're going to use all of this information to figure out how many barrels are produced.

00:40 First, we're going to figure out how many barrels are produced in the first 35 days.

00:43 So we're going to do first.

00:45 We're going to integrate this.

00:47 We're going to integrate the first part of our piecewise function, which is 800.

00:51 So we get 800 t.

00:57 So we get 30 times 800 minus 800 times zero.

01:06 So we get 24.

01:08 We do the exact same thing over the course of 30 to 35 for our second equation, which is 2 ,600 minus 60 t, which it becomes 2 ,600 t minus 30 t squared.

01:31 That should be 30, that should be 35.

01:34 And if we calculate this out, we get 2 ,600 times 35 minus 30 times 35.

01:44 Squared minus 2 ,600 times 30 minus 30 times 30 squared.

01:56 If we calculate all of this out, we get 2 ,600 times 35 is 91 ,000 minus 30 times 35 squared becomes 54 ,250.

02:23 And if we subtract the other side, we get 51 ,000, which then becomes 3 ,250.

02:41 So we add this and this together, and we get 27 ,250 barrels...

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