Textbook Question
9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.
a. The displacement between t=0 and t=5
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.1.45a
Oil production An oil refinery produces oil at a variable rate given by Q'(t) = <1x3 matrix>, where is measured in days and is measured in barrels.
a. How many barrels are produced in the first 35 days?
Verified step by step guidance1
Identify the given rate of oil production function, which is the derivative of the quantity produced, denoted as \(Q'(t)\). This function represents the rate of change of barrels produced per day.
To find the total barrels produced in the first 35 days, set up the definite integral of the rate function \(Q'(t)\) from \(t=0\) to \(t=35\). This integral will give the total quantity produced over that time interval.
Write the integral as \(\int_0^{35} Q'(t) \, dt\). This represents the accumulation of production from day 0 to day 35.
Evaluate the integral by finding the antiderivative \(Q(t)\) of \(Q'(t)\), then compute \(Q(35) - Q(0)\) to find the total barrels produced in the first 35 days.
Interpret the result as the total number of barrels produced during the first 35 days, based on the integral of the production rate.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative as a Rate of Change
The derivative Q'(t) represents the instantaneous rate of oil production at time t, measured in barrels per day. Understanding that the derivative gives the rate at which the quantity changes is essential to relate production rate to total production.
Recommended video:
04:16
Intro To Related Rates
Definite Integral for Accumulated Quantity
The total barrels produced over a time interval is found by integrating the rate function Q'(t) over that interval. The definite integral sums the instantaneous rates to give the accumulated quantity produced between two time points.
Recommended video:
05:43Definition of the Definite Integral
Evaluating Definite Integrals
To find the total production in the first 35 days, you must evaluate the definite integral of Q'(t) from t=0 to t=35. This involves finding the antiderivative of Q'(t) and computing the difference of its values at the limits.
Recommended video:
05:43Definition of the Definite Integral
Related Practice
Textbook Question
Consider a solid whose base is the region in the first quadrant bounded by the curve y=√3−x and the line x=2, and whose cross sections through the solid perpendicular to the x-axis are squares.
a. Find an expression for the area A(x) of a cross section of the solid at a point x in [0, 2].
Textbook Question
17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.
a. Determine the position function, for t≥0, using the antiderivative method
v(t) = 9−t² on [0, 4]; s(0)=−2
1
views
Textbook Question
9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.
a. The displacement between t=0 and t=5
Textbook Question
A surface is generated by revolving the line f(x)=2−x, for 0≤x≤2, about the x-axis. Find the area of the resulting surface in the following ways.
a. Using calculus
Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
a. The area of the region bounded by y=x and x=y^2 can be found only by integrating with respect to x.
1
views