Page 3: Area Between Curves and Tunnel Construction

This page continues with solutions for the area between curves problem and begins addressing the tunnel construction application, providing excellent examples for Anwendungsaufgaben Integralrechnung mit Lösung.

The solution for the area between f(x) = x² - 6x + 10 and g(x) = x is presented in detail. It includes finding the intersection points of the two functions, which determine the integration limits. The solution demonstrates how to set up the integral for the area difference between the two curves.

Example: To find the area between two curves, integrate the difference of the upper function minus the lower function between the intersection points.

The page then transitions to the tunnel construction problem. It begins by establishing the equation for the outer boundary of the tunnel, given the inner boundary equation. This problem showcases the application of integral calculus in engineering contexts.

Highlight: This problem is an excellent example of Integralrechnung im Sachzusammenhang, demonstrating how mathematical concepts are applied in real-world engineering scenarios.

The solution process includes setting up a system of equations to determine the coefficients of the outer boundary function. This step-by-step approach is particularly useful for students working on Integralrechnung Aufgaben mit Lösung Klasse 11 PDF.

Page 4: Completion of Tunnel Construction Problem

This page completes the solution to the tunnel construction problem, providing a comprehensive example of Fläche zwischen zwei Graphen Aufgaben pdf. The solution demonstrates how to calculate the volume of concrete needed for the tunnel construction.

The process involves calculating the area between the inner and outer boundary curves of the tunnel cross-section. This is done by integrating the difference between the two functions over the width of the tunnel.

Example: The volume of concrete is calculated by multiplying the area of the cross-section by the length of the tunnel: Volume = Area × Length.

The solution includes detailed steps for setting up and evaluating the definite integral. It also shows how to interpret the result in the context of the problem, converting the calculated area to volume.

Highlight: This problem excellently demonstrates the practical application of integral calculus in civil engineering, making it a valuable example for Integralrechnung im Sachzusammenhang.

The final answer is presented clearly, stating that 30 m³ of concrete is needed for the tunnel construction. This clear presentation of results is crucial for Textaufgaben Integralrechnung.

Page 5: Wolf Population Growth Problem

This page focuses on the wolf population growth problem, providing an excellent example of Integralrechnung Anwendungsaufgaben PDF in a biological context. The problem demonstrates how integral calculus can be applied to population dynamics.

The solution begins by integrating the given growth rate function to find the population function. This step illustrates the fundamental relationship between a rate of change (derivative) and the quantity itself (integral).

Definition: In population dynamics, integrating the growth rate function gives the change in population over time.

The first part of the problem asks to calculate the increase in wolf population over the first ten years. This is solved by evaluating the definite integral of the growth rate function from t = 0 to t = 10.

Example: The increase in population is calculated as: Increase = ∫₀¹⁰ w'(t) dt, where w'(t) is the growth rate function.

The second part of the problem involves working backwards to determine the initial population. This is done by using the given information that after 20 years, the population is 172 wolves.

Highlight: This problem showcases how integral calculus can be applied in ecological studies, making it a valuable example for Integralrechnung im Sachzusammenhang.

The solution concludes by stating that the initial population was 16 wolves. This clear presentation of results is crucial for Integralrechnung Aufgaben mit Lösung Klasse 12.