The graph below shows the rate in gallons per hour at which oil is leaking out of a tank. 40 30 gals/hour y = r(t) 20 10 1 a) Write a definite integral that represents the total amount of oil that leaks out in the first hour. b) Shade the region whose area represents the total amount of oil that leaks out in the first hour. c) Give a lower and upper estimate of the total amount of oil that leaks out in the first hour.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![### Educational Content on Calculus and Definite Integrals
#### Problems and Exercises
**Problem 3:**
The function \( F(x) = x \ln x \) has the derivative \( F'(x) = 1 + \ln x \).
- **Task:** Compute \(\int_{1}^{e} (1 + \ln x) \, dx\)
- **Method:**
- **a)** Using left and right sums with 50 subdivisions.
- **b)** Using the Fundamental Theorem of Calculus.
**Problem 4:**
- The graph below shows the rate in gallons per hour at which oil is leaking out of a tank.
![Graph Description:]
- The graph is a curve \( y = f(t) \) plotted with time on the x-axis and rate (in gallons/hour) on the y-axis.
- The x-axis shows hours denoted as 0, 20, 30.
- The y-axis is marked at intervals of 10, showing values 10, 20, 30.
- **Tasks:**
- **a)** Write a definite integral that represents the total amount of oil that leaks out in the first hour.
- **b)** Shade the region whose area represents the total amount of oil that leaks out in the first hour.
- **c)** Give a lower and upper estimate of the total amount of oil that leaks out in the first hour.
**Problem 5:**
- **Scenario:** After \( t \) hours, a population of bacteria is growing at a rate of \( 2^t \) hundred bacteria per hour.
- **Tasks:**
- **a)** Write a definite integral that measures the total increase in the bacteria population during the first 4 hours.
- **b)** Evaluate the definite integral in part a.
**Problem 6:**
- **Context:** The rate of production for a chemical follows a given rate that requires further calculations.
#### Graphical Explanation
The graph provided shows a rate of leakage defined by \( y = f(t) \).
- **Axes:**
- X-axis: Time in hours.
- Y-axis: Rate of leakage in gallons per hour.
This representation helps visualize and compute the total leakage over a specific duration by considering the area under the curve. By applying calculus techniques, such as the Fundamental Theorem of Calculus and Riemann sums, one can understand changes](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d5a9cae-3c86-4153-9ed8-d2e45fd1e238%2F710d1d57-125e-497f-bfaf-8bb4fcdc7fe2%2Feo7ju9_processed.jpeg&w=3840&q=75)
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