2021 Live Review 4 | AP Calculus AB | Integrals & the Fundamental Theorem of Calculus

Mark Corelli starts the video by addressing internet connectivity problems in Mississippi and welcomes everyone to the live session. He mentions that Verge will join shortly and proceeds to begin the slideshow. Mark talks about the day's agenda, which includes working with integrals, anti-derivative rules, definite integrals, and the fundamental theorem of calculus. He also discusses the importance of pausing the video to access forms and resources, and hints at providing annotated solutions and extra material in the future.

The warm-up problem involves finding the definite integral of a piecewise function from one to five. Mark emphasizes understanding the concept of a definite integral as the area under the curve. He uses a low-tech graph to illustrate the process of finding the area, which involves adding the areas of two triangles and a rectangle. Mark also addresses feedback from viewers, reassuring them that calculus is challenging and commending their efforts.

Mark moves on to discuss indefinite integrals, using the example of the integral of 2e^(x/2)dx. He explains the process of using a u-substitution to solve the integral, transforming the problem into a more manageable form and then anti-differentiating to find the family of functions that represent the solution. Mark also differentiates between indefinite and definite integrals and emphasizes the importance of showing work to avoid choosing the wrong answer.

The discussion continues with definite integrals, specifically focusing on a piecewise function defined by g(x). Mark breaks down the integral from negative 1 to 3 into two parts, using the properties of definite integrals to separate the calculation into intervals. He calculates the area under the curve for each part, considering the geometric interpretation of the function and its continuity. Mark also touches on the concept of removable discontinuities in the context of definite integrals.

Mark tackles a problem involving a table of values representing the rate at which people leave a stadium after a game. He uses a left Riemann sum to approximate the number of people leaving the stadium during the first 15 minutes. Mark emphasizes the importance of understanding different approximation methods and the need to show work, especially in the context of the AP exam.

The video features a problem that requires the application of the fundamental theorem of calculus to find g(7), given g(0) = 2π and a graph of g'(x). Mark uses the accumulation model to set up the integral from 0 to 7 and finds the area under the curve, which includes a semi-circle and a line segment. He highlights the importance of practice and understanding the fundamental theorem for solving such problems.

Mark discusses an indefinite integral that appears to require a u-substitution but instead leads to completing the square. He demonstrates how to rearrange the integral to find a perfect square trinomial, which factors and leads to an inverse tangent function. Mark emphasizes the importance of understanding the relationship between derivatives and integrals, and suggests differentiating the answer to check work as a strategy.

Mark attempts to solve an indefinite integral using a u-substitution but realizes it's not the correct approach. He discusses the importance of trying different methods and learning from mistakes. The video also touches on the idea of factoring the denominator and the concept of completing the square in the context of integration. Mark highlights the use of trigonometric functions and the potential for interesting outcomes in integration problems.

Mark calculates the area under the curve of a function k, which consists of two line segments and a semi-circle. He defines a new function h(x) as the integral from 1 to x of k(t) dt and calculates h(1), h(4), and h(-5) by finding the corresponding areas under the curve. Mark also discusses the concept of a removable discontinuity and its impact on the area calculations.

Mark differentiates the function h(x) to find h'(x), which is equal to k(x) according to the fundamental theorem of calculus. He calculates h'(1), h'(-2), and h'(-4) by evaluating the height of the graph k at these points. Mark also discusses the importance of showing work on the AP exam, especially when it comes to earning points for each part of a question.

Mark calculates the value of the integral from 7 to 9 of k(x) dx, using the given definite integral from negative 2 to 9 of k(x) dx. He uses the graph to find the area under the curve between 7 and 9 by subtracting known areas from the total area. Mark emphasizes the importance of showing all work, especially on the free-response section of the AP exam, to ensure full credit.

Mark and Verge summarize the key takeaways from the video, emphasizing the concept of integrals as areas and the importance of understanding the fundamental theorem of calculus. They also provide advice for students on how to use the weekend effectively to study, mentioning the availability of practice problems, answers, and additional study materials. Mark and Verge thank the viewers for their participation and encourage them to continue engaging with the content.